Now ref # 12 is the S.A.M. unit, and ref #4 is for the coil, and note that there are 2, if there is a "kit", I have not come across one.
Also, to say to replace the S.A.M. unit would be kind of wreckless without trying to find out what the problem is.
But this system is unique, in that the SAM unit requires a minimum voltage to run, otherwise the coils will shut down, so you should get your hands on a multi-meter, and sit down and plan on putting some time in on figuring this out, someone here probably has a link to the Kohler mechanic's manuals, maybe they will put up the link.
Or you could go the other route, and replace parts that everyone suggests, which would require a big checkbook.
Or you could take it to a dealer, which would require bigger checkbook.
I suggested the quick look for corroded keyswitch terminals, because it is a common problem spot, and speeds up the process a bit.
But since a voltage drop could easily explain things, either by cutting power to the SAM unit or else the fuel shutoff solenoid, I would suggest the first thing that you do is a simple voltage test of your battery running, and check the alt output, and at the regulator, just to start things....