It's not necessarily recommended, but you can turn on a computer by touching something metal to the pins for the power switch on the motherboard. Do you know what model your motherboard is? If you do, I can post a link to a diagram.
You can "hardstart" it by for example placing a screwdriver tip on the connector pins from where the power comes from when you press the button on your computer. If that does work= 1 If that doesnt work=2
1. Good, that ruled out something being wrong. Most likely one of the wires isint secured tightly or have come loose. Secure them and try again.
2. Not so good, try reseting the CMOS battery. (a round battery often near the middle of the mobo) by removing it for 15 min and then reseating it. BUT this will reset the bios and the motherboard will go back to factory settings. But your data and all should be 100% fine.