On my 2002 Mini, the handbrake does not affect the right rear tire. The problem is fairly urgent, as it was only discovered by the people doing the mandated regular checkup, thus making non-essential driving illegal.

The cable appeared stuck (the spring being in the always on position when I looked at it) so replacing that seemed the obvious choice. The new cable, while moving more than the old one, did not fix the actual problem though. I do know that replacing the entire brake assembly would in all likelihood fix the problem, but am reluctant to do so as despite the handbrake's problems, the rear brakes all work perfectly well. However, I can't find any in-depth guides to the part between the handbrake cable and the actual brake assembly (I might not have known the right term to search for, English being a third language, at least technically, for me)

Can I fix this problem without buying new rear brakes (one new brake and one old one probably also being imbalanced enough to become a problem)?

Update:

I tried replacing the brake pads as per the guide provided by the Pelican Parts site and in doing so discovered that the torx bolt holding the brake disk in place has broken. The previous owner appears to have tried to remove it via drill, but stopped halfway through because his drilling was off-centre. The pads themselves were still in working condition and the disk appears not to be too worn.

After putting the brakes back together, still with the old pads as those seemed not to be source of the problem, the situation was as before - pressing down the pedal stopped the car easily, with all the wheels blocked until the ABS kicks in. However, the handbrake has no effect even though the rest of the handbrake assembly seems to work.

The inner right rear pad was the least well off, requiring a bit of work to get it loose from its fitting and a lot more to get it back. Does the MINI handbrake only affect the inner pads? From what I know, on most cars both pads are affected by the handbrake.

Can I conceivably drill through the broken torx bolt while the brake disk is still in place?

The hexagonal bolts holding the brake disk restraints are hard to reach and even after I managed to get at them, they required far more force to unscrew than I could apply what with the lack of room back there.