Are you talking about the power button on the monitor or on the box. If it's on the box I would guess hard drive problems. If it's on the monitor just pull the power from the monitor and plug it back in.
Assuming you have not figured this out yet or tossed it, if the system is getting power to the board but not sending a signal to the monitor, you may have blown caps on the mother board. There is a string of them about 1/2 to 3/4" tall next to the processor. One or more may either be expanded or blown.
Crack the case and check them out. If they are blown, I suggest replacing the mother board, since a normal soldering gun will not get hot enough to remove the caps and allow replacement.