If your pump keeps running after you turn the car off of never runs, this video gives some cheaper options than buying a new pump. If you're happy soldering.

Run through disassembly, clean and rebuild of the electric power steering pump for Mini R50 and R53 (2000-2006).

I do NOT cover removal or installation. There some excellent guides already, like modmini
   • MINI Cooper Replace Power Steering Pu...  

Assembly of the pump itself is covered in my previous video:
   • Mini R50 R53 Rebuild Power Steering Pump  

Fuses:
Ignition fuse F39 5A in left foot well (green white wire).
Power to the pump fuse FL4 100A engine bay underside of fuse box (heavy red cable).

Some of the electronic parts:
Mosfet - 2N04H4 TO-220 = IRF3205 (TO220)) = IRF 3206
These fail causing pump to always or never run

Capacitors (all the same) 2700uF (M) 25V epcos LL B41866-S5278-M5 40/125/56
Rectifier diode schottky PBYR 4045WT PHm0419 A2 = PBYR 3045PT
Diode (ON) MR2520L
Small capacitor 47 VFC 6M1

Bearings - 6001 2RS