Unplug the system, remove or disconnect the battery and hold the power button for 30 sec. Remove the base cover and memory modules - then reinstall the memory modules.
Try powering up on AC only - no battery. If the system will power up without but not with the battery installed, the battery is bad -- replace the battery with a new one.
If it still won't power up, replace the mainboard.
You really have two options - the cost of a mainboard ($250+) or a replacement system. If the system is three years old or more, the latter makes more sense.