|If you push and hold the power switch in for 4-5 seconds it will turn off.|
When it starts up do you get any beeps? Do the numlock, caps lock, and scroll lock LED's flash? If so you may have a problem with the video adapter. If you have both on-board video and a video card try both.
If no beeps (and the damage was done by the power problems and not people fiddling inside the case) you probably have a bad motherboard and/or cpu. Maybe the power supply too. Try disconnecting all the drives and removing all the cards except video and see if anything shows on the screen.
There's really no practical way of testing the motherboard, cpu and PSU without swapping them out with known good ones and just seeing what happens.
You might try clearing the cmos with the motherboard jumper. People often think doing that is some kind of miracle cure--which it's not--but occasionally after diagnosing a motherboard as DOA we'll hear "Hey guys guess what? I cleared the cmos and it works fine now." So if that happens in your case at least we can say that was one of the recommendations.