One other thing you can try is to take both the battery and mains power out. Hold the power button down for at least 30 seconds (this should remove any remnants of power in the unit) plug the mains power back in with the battery still removed. Now try starting
Remove the battery, then plug the power cable into the computer. Try turning on the computer by pressing the power button. If the computer does not power up, more troubleshooting of the hardware will be needed.
For example, the issue could be as simple as a faulty power button, or it could be a power supply issue. Unfortunately, these types of issues will require that you swap out potential parts until you find the culprit.