"Doesn't turn on" can sadly be anything, but I've seen a couple times that this can be literally just down to a faulty power button wiring. In the unlikely event that it is, it's a very easy workaround.
Look inside, and follow the cable(s) from the power button to the two pins on the motherboard. These two pins should be directly next to each other (and are often labelled on the board as PWR_BTN or similar) - gently remove the power button wires (and remember to put them back on the same way afterwards), then try shorting the two pins with anything metal, I usually use just a flat-head screwdriver.
I repeat, it's not overly likely, but in case it is the power button itself that's faulty, the computer should then turn on as if you pressed the power button. Otherwise, proceed per the other post.