This?
It's been a bit since I had mine apart, but if I remember correctly the button just pushed on a shaft which pushed that little contact arm in the center of the picture inwards (towards the steering column) to the little contact post.
If mine was giving me connection problems I think I would probably just take the contacts off and sand them down with some fine sand paper then figure out some coating that would keep it from oxidizing? (although I would probably get lazy and just lube it up real well with dielectric grease)
Maybe redoing the solder joints as well if they looked like 'cold' joints
I don't even know if my washer pump works, I haven't ever used it. I am a big rainx fan so I don't generally use the wipers to clear the wind screen