The melted wires makes for a scary start to this situation. I would first make sure the CDI unit is getting power. Also be sure the kill switch and the ignition key switch are making good contact. Another thing that can happen to Kawasaki’s of this vintage is poor contact when turning on the ignition. It helps if you turn the key very fast & forceful from off to on.
Do you have a wiring diagram? You will probably need this to track down the problem.
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