The lights work now although it turned out (embarassingly) that the switch was not the problem.
After I installed the switch the brake lights still didn't work. I fiddled with the fuse and suddenly they worked. When I put the fuse cover back on - they stopped working again
I cleaned up all the fuse connectors and replaced all of the fuses and that seems to have fixed the problem.
Still, now I have a spare switch in my glovebox
So I can probably get away without bleeding the brakes.
Also I got new pads not long ago and they seem to be very noisy. What pads do you use and recommend???