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This is a fairly common problem on these coopers, there is an electronic controller that is part of the power steering pump which fails. The repair is to replace the pump/controller assembly.
The pump has the control unit electronics built onto the side of it, the pump is supplied with constant power from the main battery feed, the electronics do the switching and controlling when to power it up - these pumps can either fail from water intrusion, or just general electronics failure. The pump is designed to keep running if you stall the engine, so you don't loose steering, but sometimes when they fail they just run all the time.
The part from the dealer is around $700, labor is about 1.5 hrs.