This might be a uncommon problem which may prevent computer to startup. It’s better to check up the power button before going into deeper troubleshooting process.
The power button can be checked by opening the computer case and shorting the pins in which the wire from the power button is attached. Before starting, keep your mother manual as this comes in handy to find out the power button pin.
Important: Please be aware of the Problems of Electrostatic Discharge (ESD) to the internal components of your computer and steps to prevent it.
Follow these steps to turn ON computer without the power button.
1. Open the computer case by removing screws from the back side of the case.
2. Locate the wire coming from the power button ( Front panel wires). You could identify these wires as a bunch of wires coming from the inside of the front panel.
3. Locate the pins where the front panel wires end.
4. Pull out the wire that comes from the power button. This can be done by looking the motherboard manual. This wire should end up connecting two pins in the mother board. Some mother boards have a separate button for powering up the PC as shown in the image.
5. Take a screw driver or any metal piece, touch both the pins and check whether computer turns ON. If you have a faulty power button, this step should power ON your computer.
Be patience and take ample time to study your mother board manual and to find out the correct wire that comes from your power button. Or else you might end up shorting wrong pins which may or may not cause undesirable output.
You may be needed to change the power button if it fails the test. But it may not be possible to switch ON the computer every time by short circuit method. In the mean time one can use RESET button as a replacement for power button.