Firstly as said you need to find whether the switch is at fault or whether its something else in the circuit.

You will find the fan plug on the radiator just below the top hose, bridge that with wire.

The fan will take quite a while to kick in at idle however.....does it actually over heat?

As for the coolant being brown, its obviously been run with water in for a period of time, drain the system, run a hose pipe inside to flush everything out and then put in anti-freeze, top up to level with water and run the car to temperature with the cap off.