The inner parts of the small switch are corroded. Pressing the button has no effect – the corroded parts don't conduct electricity.
Blowing into the button and brings moisture in there → Bang. Switch works… until it dries up again. But there is one problem: Saliva is corrosive as well. In the long run it will destroy the small metal plates and – if you blow like mad into the DS other parts may corrode. Seriously: Don't do → that. Yes, it is worth reading the initial post of this thread.
Ok, I'll keep that in mind.
If the problem persists, I'll try getting a brand new replacement switch or ribbon cable assembly thingy.