cass,
Is your plug wet or dry after you try to start it multiple times? It is possible that it is the CDI. Sometimes they fail weird before they die outright. I once had one go (non KX) under your circumstances - after a rebuild. As I was breaking it in on a slow trail ride the bike began seizing repeatedly. It took about 7 or 8 seizures before I got back to the truck then it wouldn't start, which I attributed to the multiple seizures. In my shop I found no spark then tested the cdi which proved to be bad. Last thing I suspected - I was expecting an air leak.
If you are putting gas down the plug hole and it still doens't fire then there is either, no spark, no compression, or spark at wrong time. Those are the only 3 possibilities. Sounds like you have ruled out the first two so is has to be the third, "spark at wrong time". That can only be sheared flywheel key, damaged impulse pickup, or bad CDI. You have ruled out one of these so it could be, "damaged impulse pickup" or "bad CDI". An ohm meter should verify the pickup (you might have already done this) - you can have heat issues with pickup failures but that is not the case here. Sounds like "bad CD" is the winner.
Heres some suggestions:
1. CDIs arn't cheap. It would be nice to borrow a cdi and try it -but hard to find.
2. Try kicking it over with a timing light on the flywheel to see if it fires at the mark.
Hope this helps. Cam.