Hard to say if that is the correct part. realoem.com says that is the actuator/air recycle part. It could be the actual switch or that part. It looks like that part is in the left/NS (on UK car). Prise off the footwell cover with a plastic tool and see if you can see it, you may have to stand on your head! Quite awkward in there... if not, you are going to have to start taking the dash to bits, there are guides for that but I could find nothing relating to the actuator replacement for a MINI, same as you. Plenty about the fan, nothing about this part.
Voltmeter might come in handy too but I would start with that part since it is quite cheap second hand. You could try taking out the old one first before buying to see if there is an easy fix.
Cooper 2009, R57.