It sounds like the brake arm side of the hub is the problem as well as the culprit of your wobbly issues.
I would not recommend messing around with the right side bearing cone and thin lock nut as that side is not what spun when the wire job failed. The brake arm side spun and now it needs to be re-set back to where it was previously to correct the problem. It too is the bearing cone for the left side of the hub bearings.
It has spun away from the hub and now the wheel bearings have too much free play.
The hub rebuilding link you have should serve you well to getting things back in order.
Here's another one specific to the CB110e hub that may help you as well:
Park Tool Co. ParkTool Blog Coaster Hub Overhaul (Pedal Brake Hub)
And a very good read on coaster brake hubs by Sheldon Brown (RIP).
Pay close attention to the reaction arm section, particularly where they talk about mounting it securely to the chain stay. It must not be able to move up or down under any circumstances, EVER.
One-Speed Bicycle Coaster Brakes