Random suggestions:
Replace the spark plug, even if it is relatively new. I once had a plug "cold-fouled" by the junk in today's fuel. It would spark in air but run to ground under compression.
Remove the air filter and carb linkage. Close the choke by hand and open the throttle plate about 1/3. The Kohler is very cold natured and won't start unless the choke is completely closed. If it starts and runs, the choke adjustment is the problem.
If it still won't start, while the air filter is off, open the choke and spray a shot of starting fluid. If it starts with help, the carb rebuild is suspect. Does it still have a gas filter? Replace it just to be safe.
Have you replaced the air filter? I once saw one so plugged, it wasn't passing any air.
jack vines