i'm not familiar with Kohlers, but usually the switch has an an earth / metal plate on it that earths the spark out when turning the switch to stop position, therefore maybe the part on the switch isn't touching the engine / cowling (wherever it's located)
All's a switch does on a petrol is take the spark away from the plug to stop the engine, thinking of this rule may help you with your problem.
Hope this helps?
in the meantime you could cut the earth from the switch and just touch the engine block while running it, then connect back up when you fix the problem at a later date? or maybe you could run a wire directly from the off position on your switch straight to your engine block or ground?