Get a generic can of 'carb cleaner' from an auto parts store, or Walmart.
Remove the air cleaner cover and remove the air cleaner filter.
Locate the carb inlet, insure the choke on the throttle is OFF.(choke plate is parallel to the carb throat).
With the red straw on the carb cleaner can, spritz a few shots of carb cleaner into the carb inlet.
Sit on the mower, open throttle to half, and crank engine.
It should fire and run for a few seconds then die.
If it fires, locate the solenoid on the bottom of the carb float bowl with two wires going to it. Remove the connector with two wires.
CAREFULLY, using two jumper wires and two opened paper clips, stick the negative into one of the slots on the solenoid bottom, then touch the positive wire and paper clip to the other slot. You should hear and feel a 'click' as the solenoid coil pulls the slug down and allows fuel to flow into the jet.
If you hear the 'click' when you jumper the solenoid, you will need to have the carb taken off and troubleshoot no fuel. If you do not hear the click, the solenoid has gone bad, and will need to be replaced. Some can be field replaced, and some require a carb replacement.
However, what I do now is I remove the solenoid, pinch off the fuel line, find a short bolt(or make one) with the proper thread as the solenoid thread and a copper washer and leave the solenoid off, and plug the threaded hole with the short bolt and copper washer. However, it appears Kohler embeds the main fuel jet into the solenoid, so it can't just be plugged.
edit to add: This may be your solenoid with only one wire:
https://www.walmart.com/ip/Carbureto...6&gclsrc=3p.ds
If so, just touch the positive batt terminal to the red wire of the solenoid and listen for the 'click' sound. This may also be why your batt is running down, if that solenoid is live all the time. fiddle with it, and see if you can make it click and stop clicking by touching the red wire to the batt plus term.