I have a Kohler kt745 26 hp. Hard to start but runs good otherwise. The only way I can get it to fire is hook it up to a battery charger set on 50 amps or start setting. The battery itself does seem to turn the engine over fine without the charger but won't fire. I figure it must be a weak coil. Just thought I check with you guys.
Before blaming the coil I would make sure that the choke is fully closing when you put the choke control in the closed position. Not uncommon for the cable to move, not allowing the choke to close fully.
