A kickstart lever that slips or does not engage is unrelated to compression or lack of it.
KIPS systems are notorious for failure, especially in earlier (< H) models. Yours is a C3 model, does indeed have a KIPS, and is air cooled (no water jacket).
If you're mechanically inclined, figure out how to check your KIPS system. Basically, you'll want to remove the pipe and while looking up into the exhaust port, manually activate the KIPS (probably by manually manipulating the activation rod on the RH side of the engine). You should be able to see the subport drums on either side of the main port rotate in sync and return to an 'at rest' state in sync. Also, the mail exhaust valve (a flap in the middle of the port) should raise upon activation, return to a lower position when 'at rest'.
Go to:
http://www.buykawasaki.com
put in your model and year to see an illustrated parts breakdown for your bike. 'Cylinder' shows the KIPS, 'Engine Covers' shows the activation rod and such.
Kinda late, but with the carb off would have been a great time to take the reeds out and check them for condition. They should seat on the cage and not be at all frayed, chipped or cracked.
My KIPS has not ever broken. Generally, people that know (CDave) say that a non-responsive bottom end will be the result of a busted KIPS. Maybe that's because they tend to stick in the 'open' state as opposed to the 'closed' state.
In any case, it needs to be fixed if broke.
Sounds as if you've covered the simpler points.
Now the fun begins, 'eh??
Good luck. Keep us arm-chair-mechanics advised!