Computer won't power on

One of 2 things come to mind. Either the power supply if fried or your button is busted. You can check the buttom by opening you units case and removing the front panel. Be careful removing as to not bust the tabs that hold it in place.
Now the start button could be busted either of 2 parts, the front panel or the actual button wired to the unit. Check to see that the nub from the panel is busted or not. If not then your problem may very well be the button wired to the unit.
I don't know the innards to your unit but the button may just sit in place with plastic nubs or screwed in.Make sure you have a back up Button ready before removing as to remember where it wires to exactly.
Most experts will tell the average user not to turn on a unit without placing all panels back in place but that's hog wash if you ask me. Do make sure all wires are snug though. I don't replace the cover until I know it works. (unless it is the panel button. Don't attempt to unplug or plug-in any wires while the unit is on.
I have shorted units by doing doing so but then I was distracted and didn't realize at the moment. Anyway if switching the button for a new one does not work, your power supply is then most likely no good. Unplug make sure to get same exact size with same hook ups sometimes best go with the same model brand etc. Remember the wire placing.
Removing the power supply can sometimes be a little tricky. To start there should be 4 screw on the outside back. Inside may or may not have any screws. Power supplies can run some money cheapest I have seen Is about $60 or so new. A start button should only be a few $. May check with repair shops to see if they have a usable used one.