After removing the plastic cover, disassemble the power button by removing the screw which holds it to the monitor chassis. You will have the power button board with three wires: red, brown and black.

The wiring of the power button is simple. The button simply takes the red wire and put it temporarily to ground through the switch, while the brown wire act as the +V for the little LED (which has the same ground wire as the button).

If you just want to fix the button, the order of the wires is the same as the first photo: with the board on the soldering side, starting from the screw hole, red-brown-black