So you do have the Smart Spark, which have been known to be a lot of trouble. A member here named Rivets probably has the Kohler Manual on speed dial, maybe he could throw up the link, if not I'll dig around for it.
The most common problems when I have SmartSpark troubles, is not the SmartSpark box, or S.A.M. unit, but corroded wiring connections somewhere in the system. The dealer will quickly sell you the new S.A.M. unit, but they are non-returnable, and they cost $100ish.......
Just to speed things up, pull and look at the terminals on the back of the keyswitch for corrosion, as I have found that to be the petri dish for SmartSpark
trouble, as the unit needs voltage to the box, and it usually comes directly from here, and poor corroded connections make for some squirrelly symptoms.