The power button on the front of the computer is a momentary switch, this switch is like the old doorbells that would ring as long as the button was pressed. This switch when pushed shorts between two contacts on the motherboard which initiates the PSU. What you have described could be either a bad momentary switch, or a problem with the motherboard.
One method would involve using a multimeter set to the Ohm scale to test the continuity of the momentary switch or short the two header pins on the motherboard which the wires from the switch are attached. If the meter zeros out the switch is good. If when you short the two pins and the PSU doesn't start then the problem is with the motherboard. This procedure will require the PSU being plugged in order to test the shorting portion.
If you are not comfortable with this method there is another which if the instructions are followed will be safer and will determine if there is a fault with either the switch or the motherboard. The instructions are below.
No matter which method is used you need to be aware that static electricity in your body can damage electronic components inside the computer. Before touching anything inside touch the metal of the case to discharge any static electricity that is stored in your body. You also need to be aware that with the first method there will be live voltages present during the short test. The purpose of this procedure is to bypass the motherboard to test the PSU.When a computer begins the boot process the motherboard initiates the start up of the PSU. Because of this it is difficult to determine whether the problem is with the motherboard or the PSU when a computer shows no signs of starting up. The purpose of the procedure is to determine if the problem is with the motherboard or the PSU. For safety purposes please follow the instructions step by step.This test is for ATX PSUs. Some manufacturers use non-ATX PSUs with 20/24 pin connectors that do not have the same pinout as a ATX PSU.
First, shutdown your computer. Then unplug the power cable going into your computer.
Once you have opened the case, touch the metal of the case to discharge any static electricity.
The connector of the PSU which connects to the motherboard is readily recognizable by the number of wires in the bundle. To disconnect it you will need to press on the plastic clip to disengage it and then pull the connector up and away from the motherboard. Please take notice of the location of the locking tab and the notch on the socket of the motherboard, this will only connect one way as it is keyed. This wire bundle will have a memory of the way it has been installed and will want to bend back that direction, you may have to play around with it to find a position that the connector will stay in the same position while you run the test.
From the top left to right the pins are 13-24, the bottom from left to right are 1-12.
Please notice that there are PSUs with 24 pin and 20 pin connectors, the location of the green wire in the 24 pin connector is #16, and the green wire in the 20 pin connector is #14. If you look at the connector with socket side facing you and the clip on the top the number one pin will be on the bottom left corner. This makes the pin out for the 24 pin connector from left to right 13-24 on top, and 1-12 on the bottom. The pin out for the 20 pin connector from left to right is 11-20 on top , and 1-10 on the bottom. If you look at the connectors you notice that these are sockets that fit over the pins on the motherboard where the PSU cable attaches, this is where you will place the jumper. For a jumper you will need a piece of solid wire about the size of a paper clip (20-22 awg), preferably a wire with insulation. It will need to be large enough to fit firmly into the socket so that it will not need to be held in place while testing. You are at risk of electrical shock if you are holding the jumper when you power up the PSU. Insert one end of the jumper into the socket of the Green wire, and insert the other end into the socket of any Black wire.
Once the jumper is in place plug the cord back in. If the PSU is working properly the case fans, optical drives, hdds, and LEDs should power up and remain on. I would suggest that you not leave this connected any longer than is necessary for safety purposes.