I managed to find and resolve the issue.
The problem was with the switch assembly. The plastic casing broke, making it impossible to press the buttons.
I had a second light switch assembly on hand, because I installed the fog lights (I bought the one which broke from E-bay). I planned to replace the insides of the original assembly with the E-bay one, however to my surprise every part, except for that tiny plastic switch and where it attaches were identical.
So what I did was to remove the tiny plastic switch and move it to the original assembly. This was easy, all I had to do is pry off the button, remove the big transparent plastic from the inside and slide out the button mechanism.
Problem solved.