Broken power button, unable to turn onw

Have you tried to connect it to the charger? That works for most phones to wake it up or, when the battery is empty, to start the phone when reaching a certain charge.

If that does not work you could short-circuit the power button: To do this you need to do some disassembling. I am assuming that the golden plate with the two contacts (just below the blue and red wires on the right side of this video) is the contact for the button to the motherboard. If you peel this off from the other side, re-insert the motherboard, battery, and temporarily connect the display you might be able to short-circuit that contact on the motherboard. But those are some "IFs" there.

Try the charger first ;)