Your ignition switch is another matter.. It has several wires, probably 4.
Two of the wires connect the ignition and allow the engine to run. The other 2 wires are just like the KILL switch wires and, when you turn the IGNITION switch off, the engine dies.
Fix that broken kill switch. i think it's at the root of the troubles.
Im not positive about this but it's a common way the these bikes are wired. Get a manual and a wiring diagam for your bike and study it.