I put the red multimeter lead into the red on the switch with and without the key on. It doesn't give me a steady reading and jumps around from .1 to 3.1 Should it be 12.5 like the solenoid?
Please keep in mind that you are there with the mower in front of your nose and we are here with nothing but words to look at
Switch ?
which switch ?
If it is the key switch and you are testing the solid red wire on the B terminal to ground, yes it should read the same as the battery cable on the solenoid .
So a broken wire or bad fuse or bad fuse holder come to mind
Run a temporary wire from the battery or solenoid battery cable to the S ( yellow ) wire on the plug
If everything else is in the start mode, the engine should crank.
If you have some uninsulated 1/4" crimps then make a temporary by pass wire to go from the key switch plug to the solenoid battery cable
To confirm the key is working pull the solid red out of the plug and put your temporary wire in the plug, plug the key switch in and try cranking the engine .
The terminals just pull out once you have poked a tiny screwdriver in the front of the switch to depress the barb .
If you only have the crap insulated terminals then cut the insulation off push the wire through the plug, crimp on the terminal, push it onto the key switch then push the plug onto the switch.
Remember this is just a temporary diagnosis & get you out of the poo situation
Ultimately you will need to replace the entire wire including the fuse .
I find it much bette to go to an auto store & get a new fuse holder that has loose terminals for you to crimp onto your own wires .
That way you can make a new wire with the minimum number of joints to bite you on the bum latter on .