Saskia's answer above is the best solution short of replacing the part (i.e., replacing the motherboard, in most cases probably more hassle/expense than it is worth).
One quick and dirty, no-disassembly trick that worked for me several times was to rap the phone (in a bumper case) sharply against the edge of a counter with the power button facing down). Optional followup: with the button still facing down use the crevice tool (with the case off) in hopes it would catch anything that had worked lose in the thin gap between the switch and the phone body.
Not ideal but handy if you are on the road and want to give it a try.