Check the right side of the Kohler engine. There will be a spark plug on this side of the motor that is partially inserted into the spark plug hole. A thin ignition wire extends out of this hole and should connect into the rear of the spark plug. Reinsert this wire into the rear of the plug if it has come loose.
Observe the red, positive terminal wire extending out of the right side of the John Deere's battery. Trace this wire back onto the engine assembly. The wire will lead to the black, rectangular starter solenoid part. The starter solenoid features a bolt on each of its sides, and the red terminal wire must be connected to the solenoid's left bolt. Fasten the open end of the terminal wire onto this bolt, then secure it by adjusting the bolt clockwise using a socket wrench.
Ensure the black wire on the right side of the starter solenoid is connected to the solenoid's right bolt. This wire leads back and is connected to the starter motor. Fasten the open end of this wire over the right solenoid bolt as you did with the terminal wire. Secure it by turning the bolt clockwise with the socket wrench.
Find the distinctive, wheel-shaped flywheel on the top of the John Deere mower's Kohler engine. Use your wrench to remove the bolts on the flywheel's iron cover, then pull the cover off. Check the right side of the exposed John Deere flywheel. The governor's flap sits here atop a bolt. Ensure the flap is connected to the bolt. Loop the flap's wire onto the bolt and tighten it with your socket wrench as you have done with the starter solenoid terminal wire and starter motor wire.