Step 2: Fixing the electrical system
The electrical continuity problem was due to a bad wire between the starter motor and the starter solonoid (one terminal had internal corrosion), so soldering a pair of terminals to a heavy duty cable fixed that problem.
I found a "universal" ignition switch that fit the wiring harness, and that allowed me to turn the ignition "on." That solved the spark problem, although I also replaced the spark plug while I was working on the electrical system.
I was unable to use the ignition switch to engage the starter because of three heavily corroded safety switches on the mower. Rather than trying to rig up three new safety switches, I bypassed these switches and wired in a push-button starter switch. By the way, its not a great idea to bypass safety switches, but I will be the only person using this mower, and I always make sure it is out of gear when I start it, and its not powerful enough to be started with the blade engaged.
The original control panel was missing from the mower, so I fabricated one from the metal case from an old computer. Hey, I guess that counts as more recycling!