Try resetting the SMC.
If something goes pear-shaped with the power, the SMC 'trips' and cuts out the power.
Connect to power. Hold left shift, left control, left option and the power button for about 7 seconds.
Release. Wait 2 seconds. Then try powering it on.
Sometimes it naturally resets after half a year or so if the battery runs out of power.
You can also try:
unplugging the battery
press the power button so it tries to boot up for half a second. This is to drain the motherboard's capacitors fully.
Plug battery back in.
Try booting
If this works, or doesn't work, let me know so we can go from there.
Good Luck
Falcon16