First remove the battery and remove the charger. Press and hold the power button for 30 seconds. Next put the battery back in and plug it in, see if the light comes on. Press the power button to turn it on. If it still does not turn on, repeat the above steps but do not put the battery back in just plug it in and try to power it on. If this does not resolve the issue then you may need to replace the power jack.
If none of the HDMI ports work but the usb ports do It sounds like either the main board is giving out or the HDMI ports are needing fixed. For me it would be easier to change the board rather that trying to fix the ports. Have you tried to hook anything else up to any of the other ports, such as the laptop to the VGA port or anything to the components or AV ports? If any of those works, I would check for loose connections to the HDMI ports. If they are loose then either try to resolder them or replace the board.
First how old is the computer? Do you keep it plugged in all the time? The battery could be ran down or have a bad cell. I would start off by replacing the battery with a new one, or barrow one from someone that will work on just the battery swap out the batteries and see if it will then. Most likely the battery will need to be replaced.