Burnt Wire

I think you're missing the point.
That's what will happen if you don't address the 40+ amp discharge when the key is turned to start.

Putting a lamp in the battery line is a way to test for shorts with the system connected to the battery, and not melt one or more parts of the circuit.
Using a meter to check with the battery disconnected would be more effective in finding this particular problem you are facing.
Yes, I did put the lamp in the circuit, and I ordered one of those short testers last week. (should be coming in any day now.) My buddy who came and helped me thought that the broken key could have been one of the many problems and it would be best to eliminate that as a possibility. So when I fix the lock cylinder; then the last possible thing that could be shorting out is the ignition switch in the column.