Are there any electronics, circuit traces in the button, or is it just a plastic tab? I have the same monitor and if its just a plastic tab you could probably just just a pencil eraser to push the plastic. i think its just a small microswitch behind that plastic, and a led in the center to show on or off.
I would look at the back of the button that came off and see where any plastic tabs stick out, that more than likely corresponds to where you would push