Look along the right side of the John Deere mower's engine for the spark plug. Ensure the thin, black ignition wire is inserted into the spark plug. If the wire has become disconnected, insert it back into the rear of the plug.
Look at the sides of the John Deere mower's rectangular, black battery. There is one terminal wire in each side. Trace the red, positive terminal wire from the battery back to the rectangular solenoid piece. Ensure the terminal wire is connected to the bolt on the left side of the solenoid. If it is not, loop its open end over and onto the bolt, then tighten it onto the bolt with clockwise turns from your socket wrench.
Check the left side of the solenoid. The starter motor's wire leads back here and connects to a bolt. If it has become disconnected, loop the end of the starter motor wire over the bolt and tighten it with your socket wrench.
Find the flywheel. This is a unique, entirely circular iron piece that sits on top of the John Deere mower's engine block. Remove the cover on the flywheel by taking out its bolts with your socket wrench. Pull the flywheel cover off and check the right side. The governor's flap sits here and features a wire that should be connected to a bolt just beneath it. If this wire is not connected to the bolt, loop it over the bolt and tighten the bolt with your socket wrench. Position the flywheel cover back onto the flywheel and reinsert its bolts using your socket wrench.