When the LED goes off, unplug the charger and plug it back in. The cycle will repeat and each time the battery gets a few drops of power before it switches off. After doing that many times, those drops add up enough for the phone to stay on and start charging properly. I have rescued mine from a deep discharge a few times like that.

Be warned, it may take a whole day of plugging and unplugging. I can only do it at work