My daughter has a 2003 mini cooper and when she turns the key off the engine stops but the dash lights stay on as well as a sound from the engine. It is NOT the fan. If you turn the steering wheel the weird sound from the engine get's louder.

You have to turn the key off and on sometimes 20+ times before it will finally shut off.

I recorded a 54 second video of this:

screencast.com/t/kGVbpw868a

Any help would be appreciated.

 

Well by listening to the sound from the engine area, to me it sounds like the power steering motor fan. There is no real pumps in mini's just power steering motors which are electric, they have a problem with the fans. It may burn out if not updated and will take out the steering motor,cause it will get to hot. As for the car not shut down all the way, they have electric harness problems: pinched wire,water,etc. Well hope I gave you some help