Not working reverse light

I recently purchased 2003 mini cooper base that used to be an auto but the previous owner swapped to a manual. The car had been working perfectly except the reverse light doesn't work. I have traced it down to the cable not being connected to the switch on the gearbox, i just would have no idea where to start as i have found no connector that resembles the one on the switch, and i also haven't been able to find a 12v power that is only on when the car is on.