If the CH16 has the DSAI (ASAM or DSAM) ignition there will be a black plastic box on the side of the engine. That box or the coil can be bad. The 12 070 01 s is the original DSAI coil and works with the "smart spark" module. To upgrade to the MDI (magneto discharge ignition) as far as i know kohler does not have a kit for the CH16 The ones i have done I had to get the 32 707 01 s kit for a twin cylinder engine. The plug wire is shorter than the coil wire for the CH16. If you reroute it it will work.
Both of the DSAI systems are notorious for failing with intermitent spark. Also, if smart spark module loses power intermittently from bad relay contacts the engine will die like you turned the key off but will usually start right back up.