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Does it turn over normally if youy remove the spark plug? What Kohler model and spec is this?
That's what we are doing by removing the plugs. This symptom is often due to excess compression building up in the engine due to either valves being out of adjustment or by a faulty compression release. If the engine turns over easily, then we know where to turn our attention. If it still does not, then we look at either weak starter or physical binding of the engine for reasons other than compression such as a belt not de-clutching or the beginnings of seizure.
I start with the plug question because about 80% of the time it's a valve adjustment issue. I ask for the model information because not all Kohlers have adjustable valves. Some have hydraulic lifters instead, even on the same engine family, but it varies by specification. Yours are not adjustable.
Does the engine turn over freely if you turn it by hand, either from the flywheel side or from the drive pulley side? If it turns freely and you have replaced the battery already and are confident it is fully charged (verify this by jump starting from another vehicle), then
try jumping the solenoid terminals with a screwdriver to make sure you don't have a bad solenoid that's adding resistance.
Check your cables for any loosening, damage, or corrosion. Make especially sure the ground cable is getting a good clean connection. While you are jumping, you might jump from the positive battery post directly to the B+ terminal of the starter. This would eliminate all the cables and the solenoid and test the battery and starter together. However, the ground cable still needs to be verified first.
If you still don't get the engine to turn over through these methods you know you have a bad starter so long as the engine is turning freely by hand.
I am Sorry, Rick had to log off last night a few minutes before you replied.
The answer is, if you can jump the solenoid and it spins up just fine, and it will not spin up with the key switch, then you have narrowed it down to the solenoid building up a resistance that prevents full current from passing through it,
I agree with Rick that the Solenoid is likely the problem.
Please when you ACCEPT, click on Ricks ACCEPT button on His answer and not mine.
Rick is always happy to answer follow ups even after you hit Accept.