Re: Motor will not turn off
If the info pulling up is correct you have a Kohler M16 one of several possible spec numbers according to Kohler. All of the ones I looks has a solid state self contained magneto ignition systems (no points or 12v driven coil)
You need to test the kill lead at the ignition coil. Since don't a DVMM a simple analog Ohm meter will work. With the ignition switch off it should ground (reading zero ohms to ground). If it does then you have a bad ignition coil; otherwise, you have a wiring problem back to the relay.
Since you are able to start the mower, I assuming the relay, the time delay module, and the seat switch are good; although, the relay rest state contacts may be dirty.
What ever you do DO NOT apply 12 v to the coil's kill terminal as this will fry the electronics in it. To test coil's internal kill circuit to verify it working ground the kill terminal while the engine is running if it does kill the ignition then the coil is bad.