The power button is a normally open momentary switch. This is like the old doorbell buttons which would make the bell ring as long as the button was pressed. There are two wires which attach to the power switch, these are connected to two pins in the power header. This button when pressed sends a signal through the green wire in the 24 pin PSU connector which attaches to the motherboard which initiates the start of the PSU. With the computer shut down and the PSU unplugged you can test this switch in to ways. If you have a Ohm meter (continuity) you can test the switch by placing the two probes of the meter to the two wires where they attach to the power header, press the button and the meter should zero out. The other means of testing this would be to use a screw driver to short the two wire very briefly. These switches don't fail very often since they are so simple. If one does fail pressing the power button will not do anything.
As for the PSU, you can test this in a couple of different ways. One way is to bypass the motherboard to start the PSU, the computer needs to be shut down and the PSU unplugged. As I mentioned above, there is a green wire in the 24 pin PSU connector which attaches to the motherboard. If you disconnect the connector from the motherboard you can place a wire jumper between the green wire socket ( there is only one green wire ) and any black wire socket. With the jumper in place you are ready to test the PSU. Plug the PSU back into the receptacle, this will immediately start the PSU. If the PSU is working the case fan/s should start spinning and some LEDs may light as well. This indicate that the PSU is working, but it doesn't test the three main positive rail voltages +12VDC ( Yellow ), +5VDC ( Red ), +3.3VDC ( Orange ). To test these voltages you will need a multimeter with a DC Volt scale. You can read the three main rail voltages from a Molex connector or a SATA power connector which contain the black ( negative ) and the three positive voltages.
You can do this or purchase a PSU tester to determine if the rail voltages are within normal parameters.
Personally, I don't believe your problem is related to a bad switch or failed rail voltages.