OK... first test the solenoid. With the solenoid in place and the battery connected, turn the key on. Take a wire you know is hot, like the RED brake running light and tough the little soldered contact where the black wire attaches. If the bike cranks, the solenoid is GOOD as it is shutting when a trigger voltage hits it. If the starter doesn't turn over, the solenoid is bad and needs to be replaced. I would find a used solenoid that has the same general configuration as the original for ease of mounting. Many Kaw and Suz solenoids share the same manufacturer and can be used. The Kaw solenoids will often have a BLACK hanging off them where an almost Suz solenoid will not. This black wire is just a ground that would be connected to the regulator or reg/rec and you can connect the needed ground to a frame ground.
OK... the solenoid is GOOD but still no click? Start troubleshooting. Get a multimeter and test the voltage (with key on) OUT of the kill switch where it goes to the start button. If you have voltage there with the kill switch in the run position, either the button is bad or the wiring to the solenoid. With power coming off the kill switch and the key on, push the start button and test for power at the major connector for the black wire coming off the starter button. Your bike DOES have a lockout loop. To do this test, pull the black wire out of the clutch interlock switch and stick a short piece of wire in both ends of the wires that went into the switch. This will hotwire the interlock during the test. If you have power at the connector on the frame neck but not after re-plugging in the interlock, the clutch switch is bad. If you find you have power with the interlock switch actuated and the button pushed at the connector, then the problem is with the wiring in the harness between the other side of the connector and the solenoid. You can test continuity by putting the meter in Ohms scale putting the probes on the solder joint on the solenoid and corresponding pin where the trigger voltage is conveyed inside the harness side connector.