My mower stopped while cutting grass. I found a cut wire on the brake switch, repaired it, but mower will not turn over, unless solenoid is jumped, and then will not start.

You almost answered your own question. Its still the safety switch. I think even though the wire was repaired the safety switch is shot.

The safety's go through the ignition switch first for power to the solenoid and allow the ignition coil to work

Now you by passed the solenoid to spin the starter BUT the ignition coil that allows spark to the plugs is still killed because the safety switch is still telling the ign coil not to work.

Get new switch or test my theory with a jumper wire on the safety. If that works replace switch. Ignition switch and or fuse is the only other catalysts here.

PS. Don't forget to check grounding on switches and under the seat ground.