To begin, made sure your battery is good! Take 12" or so of #12 gauge or larger wire and jumper (touch to) both of large bolt terminals on the solenoid. Does the starter motor turn? If YES, problem is either ignition switch, solenoid or something in safety circuit. If NO, I'd again verify battery, check wiring leaving the battery, suspect something in the starter motor itself.
If you know/understand the mower's electrical starter circuit, and own a 12 volt test instrument (meter or light probe), troubleshooting is easy. For this type of testing I prefer the 12 volt light probe .... looks like ice pick with light bulb inside handle, one wire lead w/alligator clip extending from the handle's top. (To reach all circuit test points you may have to lengthen this lead).
Clip negative lead of meter or probe's wire to negative terminal of battery. Take meter's positive lead or tip of probe and touch battery's positive terminal .... should see meter reading or lite on probe's handle.
While HOLDING ignition key in "start" position, test for voltage at the starter. If you HAVE voltage at starter cable terminal, suspect something in the starter motor itself. If NO, still holding key in ignition mode, test for voltage at both "large bolt" (large diameter wire) terminals of starter solenoid. If you find voltage ONLY on the "battery" lead or input terminal side, but not terminal/wire going to the starter motor, problem is either ignition switch not pulling solenoid in, a bad solenoid or something making up the safety circuit.
I've seen corrosion on internal contacts of ignition switches and solenoids act as you describe. Also a loose or corroded wire plug connector will act the same way.
Back of ignition switch has two or more terminals. One terminal is for incoming 12 volts to the switch, another is "out" to the starter solenoid. Trace wiring to find these terminals. With key "off", check for voltage at the incoming terminal. YES, than turn key to "ignition" and check for voltage out to solenoid. NO .... switch is bad or something in safety circuit is "open". YES, than cycle key "off" than "ignition" ... does solenoid "click" ? NO ..... check wiring from switch to solenoid ..... or suspect bad solenoid.
Once you know the function of each switch terminal it can be further checked using a meter. If you now suspect something in the safety circuit, don't ASSUME anything. You must check and recheck wiring, switches, jumpers you've installed around these switches, any wire connections .
The connections should be as follows:--
-wire from battery + is to post on solenoid
-wire from starter is to other post on solenoid
-wire from battery - is grounded to frame
If you jump a connection between the solenoid posts the starter should turn.
