Hey Ninetoe,
Here is a handy gear ratio calculator
LINK
You just type in the number of teeth on your clutch, the number of teeth on your jackshaft input, the number of teeth on your jackshaft output, and the number of teeth on your rear sprocket. It will then give you your overall ratio.
For your Briggs engine, I would recommend a ratio of between 12:1 and 10:1, depending on how you ride and your topography. If you need to climb steep hills, then you might go more towards a 13:1 or 14:1 ratio, but 16:1 ratio would really limit top speed.
I can not really see how many teeth your different sprockets have, but looking at your pictures, it looks like you may be able to leave the rear sprocket as is and change the INPUT sprocket on the jackshaft. Making this sprocket larger would do the same as making the rear axle sprocket bigger. That output jackshaft sprocket looks pretty small, maybe 10 teeth. Then if the rear sprocket is 40 or so that gives you a 4:1 ratio in the back. If your clutch sprocket is 12, you would need a 30 tooth sprocket to get you to 10:1.
To get the overall ratio you divide rear axle sprocket by output sprocket on jackshaft, that's one ratio. Then you divide the input jackshaft sprocket by the clutch sprocket, that's the other ratio. You then multiply these two ratios to get the overall ratio. 4:1 times 2.5:1 would be 10:1. If you changed the jackshaft input sprocket to 36 in this scenario, you would get a 12:1 ratio.
Hope that helped,
Gilbert