You should provide the model and market for your Land Cruiser. Different models and markets may be different.
Have you tried disconnecting the wire harness from the switch to see if the car will start? Maybe one of the kids spilled something in the switch and your cleaning it didn't fix the problem.
There isn't much to the circuit. If the light on the dash is on all the time then the switch is always getting power to the dash light.
For both conditions, power and 2nd, the switch is connecting directly to the ECM. If the switch isn't bad then the problem may be the ECM.
This is the diagram for a 1996 US model Land Cruiser. E 9 (highlighted) is the switch and PWR (highlighted) is the dash light.
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