More work than fun

If your carb is in order, then it would seem to me like the problem is in either your CDI, your magneto, or your kill switch. The kill switches that come with these kits are kind of lame, so try disconnecting it and try starting the motor again. If the engine fires up then just stop by stalling the engine out. If that doesn't solve the problem, perhaps get a multimeter and use it to test the wires from your magneto to see if the magneto is generating enough power. If it is, then it would probably mean you'll have to change your CDI. If not then buy a new magneto.

I'm no expert compared to some other people here, but those are the things I would probably try.

simple stuff first

spray the plug hole with starting fluid to make sure you are getting fuel.
try to start it
disconnect the kill wire . see if it runs.
Get a real plug wire and boot from the parts store. see if you have spark.
either magneto or cdi was bad from factory. If you can , remove spark plug and have someone pedal with the bike on a stand . do a voltage test on magneto , then do one on the cdi.
Here is an old primer on testing it.
CDI ignition systems,trouble shooting,how do they work?
There appears to be a lot of misunderstanding and confusion about the capacitor discharge ignition systems used today on most small IC engines.The reason for the popularity of CDI systems is that they are easier to miniaturize than the old fashioned but rugged magneto setups as found for instance on the B&S lawn mowers.They also dispense with breaker contacts which will wear out,making a maintenance-free ignition system possible (at least in theory)
CDI is a successor to the magneto based ignition systems evolved a century ago by Bosch of Germany (Their logo still shows a magneto).It was a self-contained system,no batteries were required and it was used exclusively until these became common with the advent of electric starters).Magnetos were big and expensive and lost out to cheaper battery ignition,but held their own in motorcycles due to good high speed performance,They were standard in the smaller 2 strokes as flywheel magnetos/generators until CDI systems became competitive with the advent of solid state electronics
In a CDI ignition system,a storage capacitor is charged to a high voltage (a few hundred volts) by the flywheel charging coil and, at the proper moment quickly discharged into the primary winding of the ignition coil,which is actually a stepup pulse-transformer.The secondary output is a short very high amplitude pulse which fires the spark plug.All these events take place during each engine revolution.The charging coil is usually located inside the flywheel,if there is room for it, but if the flywheel doubles as a cooling fan it has to be located outside.The spark timing is controlled by a small trigger coil located in close proximity to the flywheel which generates a low voltage pulse (actually two, a positive and a negative one), as the fly wheel magnet passes by.The circuitry in the CDI unit uses this pulse to close a solid state switch (usually a thyrister) to discharge the storage capacitor into the ignition coil primary and generate the high-voltage pulse at the secondary to fire the plug.This capacitor has previously been charged to a high voltage with the high energy pulse induced in the charging coil by the flywheel magnets.All these events take place every revolution.
Both the pick up coil and the power coil have two wires connected to them, one side of each goes to ground (usually coded black),they may be grounded internally in the magneto or connected together and brought out as a single wire and grounded outside or connected to a ground connection on the CDI unit.The two live outputs (charger coil output and trigger) are connected to the CDI unit.They can be identified by disconnecting them from the CDI and checking the resistance to ground with an Ohm meter.The trigger coil has low resistance,the charger coil higher (a few hundred Ohms typically).Both outputs are ac. signals!!,the charger output typically in the range of 100 V ac. DC measurements of this signal are MEANINGLESS and deceiving !,all they measure is the degree of asymmetry of the ac output,not the actual amplitude.Even simple multimeters have an ac position.The charger output is an unhealthy signal that could kill you under the right circumstances,sit on a wet saddle and grab the bare "white lead" for kicks. while riding around.This is not all that likely of course,but the output of the charger coil should be treated with respect.
I have never set eyes on one of these Chinese 2 stroke engines,so as far as color coding goes bear with me,I have concluded that the "tormentor" "white wire" is connected to the charge coil output and the "blue wire" is connected to the trigger coil.Shorting either one to ground will disable the ignition and stop the engine,in the first case the storage cap does not get charged,in the second it does not get discharged.Opening up the connections between these outputs and the CDI unit accomplishes the same thing,I would prefer shorting to ground esp., for the low level signal from the trigger output,a long lead would be connected to the CDI trigger input and a spurious signal could conceivably blow up the input device.Shorting them to ground does not hurt anything and is safer.I would use grounding connections going back to the engine ground and weather proof the switch and I would leave the output of the charging coil alone.It is probably possible to power other devices from the output of the charging coil,I think it could power a sophisticated high voltage low current dc circuit to minimise the drain on the coil feeding an efficient power convertor to convert this high voltage (50 V plus) to 12 or 6 volts power at higher currents.
Trouble shooting hints.Caution ,if testing for spark ALLWAYS have a plug connected with the shell grounded,never ever test with an open plug lead, you may destroy the ignition coil.
1) Disconnect leads going to kill switch (shorting mode hookup!) and try starting engine (Figure out how to stop it in case it runs) .
2)The charge coil and trigger coil are fairly robust devices and not likely to fail,measure resistance to ground (disconnect from CDI),if shorted or open circuit,they are defective or leads bad.
3) Ignition coil,disconnect from CDI if possible,measure resistance to ground of primary,should be a few ohms,measure secondary resistance to ground, should be a few thousand ohms.Caution: High resistance may be due to high resistance HT lead to plug!.Not easy to check unless you can get at coil output directly.
4) CDI unit,probably implicated if other tests OK
I hope that this post has shed some light on the mysteries of CDI ignition systems

Check your wiring against the diagram below. Disconnect the kill button to make sure it is not giving you a no spark condition. Some are defective. Make sure the plug wire is screwed securely into the CDI and you have a good connection at the plug boot, the plastic part that snaps onto the plug. If you've already done these things and have access to a good ohm meter check your ignition components against the numbers I've attached. Get back to us with the results.
Magneto

Use Low Ohm-Meter Scale About 200 Ohms

1) Check Ohms between BLACK wire and WHITE wire. Reading should be around 2 Ohms

2) Check ohms between BLUE wire and WHITE wire. Should be around 300-400 ohms.

CDI

Use Ohm Meter High Scale About 200-K

1) Positive lead on BLUE wire and Negative lead on BLACK wire should read infinite (no activity)

2) Positive lead on BLACK wire and Negative lead on BLUE wire. Should be about 130-150 K-ohms

3) Positive lead on Spark Plug wire and negative lead on BLUE wire. Should read between 135-155 K-ohms

Switch to Low Scale 20K

4) Measure between Spark Plug wire and and Black lead. Should be about 2.5 - 2.7 K- ohms

With the cover off the magneto, and the piston at top dead center, the keyway (Little slot in the rotor magnet) should be at 1 o'clock, and the magnet should have the larger lobes almost horizontal. The magnet CAN be put on upside down, and you will never start that motor till it's flipped.

James Hetfield had it nearly right when he said 'Gimmie fuel, gimmie fire, gimmie that which I desire'. You need compression, you need an appropriate air/fuel mix, and you need a spark a few degrees before Top Dead Center.
Check compression: You can check this with a static pressure gage, or my feel. but the gague will give you a more accurate result. Check that your head studs are properly torqued. If you can, lap the mating surfaces of the head and cylinder flat with a mirror, plate glass, or granite surface block and 600+ grit sandpaper. A warped or out of flat head can ruin your day.
If the motor has run in the past, your magneto rotor and fuel/air should be fine. There could be a clog in the main jet, to tell if this is the case, try starting the motor with starting fluid. if it starts, your problem is carb related.
Check your plug gap! On most CDIs it should be less than .030" Gap it in to .015-.020 for now, to see if a weak spark is your problem, every CDI should be able to jump .015.
Check for a good solid ground connection to the motor. Those magneto stators can sometimes have a problem with when they're lacquered, and no good chassis ground is made. Rough up the side that contacts the crank case with a file or sandpaper, as well as where the black wire ring connector seats against the stator. Make sure you have continuity by disconnecting the CDI black wire from the Stator Black wire, and take an Ohm reading across the gap. If it reads anything besides 'open' or infinite ohms, you're good.

it sounds like the cdi is toast. will the company that sold you the motor warranty one since it is new?

I checked bikeberry and the parts have a 30 day warranty. I would bet that in good faith , the company would gladly swap you a $10 part or 2 especially since they are a sponsor here and generally recognized as one of the better quality distributors.
I would say to call them.
Toll Free: (800) 317-0479
Local: (714) 706-3400
Fax: (714) 255-1146
Hours: Mon-Fri 8:30AM-5:30PM PST

I am sure that they would not want a bad rap on these forums just because of junk parts sold to them from China.

Let me know how it goes. Your reply might sway many people as to who to buy their parts from.

one other thing......

some of the chineese manuals show to incorperate the white wire into the ignition systom.
the white wire should be cut off and thrown away. it is a generater line only! not an ignition wire. i have replaced a dozen coils for customers because of this.
sometimes it wipes out the mag too.
you can get a newcoil for about $12 and about the same for the mag.
also....
the blue wire is the HOT.... the black is ground and is usually screwed onto the mag frame
to use the block for ground
the only wire needed on the mag is the blue wire!
you can remove the white wire and instead of the black going to the mag fram rerout the black from the coil to any good ground point on the bike
the only connection that HAS to go into the mag is blue to blue

stay dry
gubba

so how do you wire the kill switch.

1 kill switch wire to each of the CD/magneto wires. What it's doing is shorting out the ignition instead of cutting it off. A simple change is getting rid of the killswitch and getting a key switch that turns to ON and spliceing it in line, best place on the blue CD/mag line.

Oh, the white wire is for a 6 volt light. Useing it straight to a light can cause problems, too, like a loss of power and lights at low RPM. Ebay has small regulator/battery pack with AA NiCads that stay charged as you ride. At night with mine I experenced no headlight drop off at stoplights and at a idle. It's like a $40 accessory but it really helps out if you have to ride at night.

I am truely at my wits end at this point. People seem to say that these kits have at least some degree or reliability but so far I have seen the opposite.[/QUOTE]

There is no comparison in reliability between the Happy Time engine and the Tanaka/Mitsubishi engines. My Mitsubishi engine sat outside for years. After changing the gas, it started up in a few pulls.

put one of the killswitch wires spliced into the blue/blue (either killswitch wire)
if your black wire is still screwed onto the mag FRAME then splice the other killswitch wire into the black/black or ground it

davec's suggestion IS the ultimate choice, no shorting to kill but an interrupt instead....... much better
or
hold your brake on and slowly let the clutch out and it will stall
CAUTION: if your carb goes into an un-controlable runaway full throttle, using the 'stall' method can rip things apart......

good luck

stay dry
gubba

Just so we're clear about this, are you starting up by peddling with the clutch disengaged and then engaging the clutch after you've gained speed? Usually you need to peddle up to 5 mph to get it started. I've been able to start my engine without disengaging the clutch but it's very difficult.

open the plug gap back up to 30.
that 7 plug might be too cold to clean itself and jump that gap. You might have fouled it although unlikely..