So, this could be a problem with your power button, power supply or motherboard. The first thing I'd do is make sure the power button plug is connected to the correct pins. If you're sure it is, try manually shorting those two pins with a screwdriver or paperclip. If the PC turns on, the power button is somehow broken. If it doesn't turn on, make
absolutely sure you're on the right pins. You can check this in your motherboard manual or sometimes it's printed directly on the motherboard near the pins. If they were the correct pins, try the "paperclip method" of testing PSUs. A tutorial can be found
here. If it works, the problem is not your PSU. If it doesn't, the PSU is the problem and it needs to be replaced.