This is a laptop, so it has a battery. Assuming there was at least some charge in the battery before this shut-down took place, there should be sufficient volts to run the computer - at least for a while.
Depending on how it is wired, the indicator light on your mains charger unit may only indicate the presence of mains electricity. Use a meter to check that there is something between about 18 and 20V DC available at the plug that connects into the laptop. If there is, great. If there isn't, this indicates either a faulty charger or a broken lead between the charger and the plug. Since the output lead on most chargers is hard-wired into the charger, this means you need a new charger.
Remove the battery and then replace it, and assuming the charger is producing an output at the plug, reconnect it to the laptop and try again.
Normally, a computer which has undergone a hard shut down using the power button like yours, will restart as far as the 'Safe mode' options screen,and, if you do nothing else, go on to boot normally after a pause of 30 seconds.
Try the above, and if it doesn't work, post back.