Toro have their technical manuals available on line for free but you will need the numbers off the ID to get the right one
In there is the wiring diagram for your mower
The modules are known for failing very regularly .
The fuse holders are also known for failures .
The ignition switch is easily verified by pulling the plug & jumping the terminals as per the wiring diagram
The solenoid inside the module rarely fails but it is electrical so subject to corrosion, badly soldered terminals, wires broken at the crimp connections etc etc etc