When I got my bike, I had a similar problem.  Had to replace the doodad that slides inside the brake handle to activate the light.  Then one of the wires broke off.

You can fix it, but do not use too much solder on it, both those contacts are close together and a little solder goes a long way.  Just take your time and it is more tedious than anything else.

I was dumb, I looked at it and said, "No problem."  Well, it was, not too much but that is fine work there.