Broken horn

My horn didn't work when I purchased my 65 coupe and I want to get it running. I removed the steering wheel and noticed that one of the wires wasn't connected to the back of the wheel. I reconnected the wire then reseated the other.

The horn worked for one beep then stopped. One of the wires is pretty ragged. Might there be a short in the wire? Should I run new wires down the shaft? If so, what guage? Thoughts?
 
It seems if the horns aren't used for a long time they do "stick". I whacked one of mine and got it working and the other I turned the adjustment screw back and forth until it worked....for awhile. Check to see if power is getting to them while somebody blows the horn; if it is, then it's the horns.