The LED should work even without the battery in your phone. Possibly your phone is not really powered off, just crashed with black screen or you have bad charger connector contacts. Try to force to reboot your phone by holding all the 3 side buttons until it reboots. Usually max 10 seconds. If that doesn't help, try to use another USB cable and charger. Better to connect to a computer USB port for testing.
Try to leave your phone on charger for at least 12 hours. If it still can't wake up, then the battery inside should be checked by a repairman. It might be overdischarged and needs to be charged by an external charger.