Try resetting it by holding down the home and sleep key until it shows the silver apple and let it reboot. THen turn it off and let it charge powered down. If it doesn't resolve itself and charge after a few hours, then totally discharging it may have messed up the battery. You may need to take it to an apple store or where he bought it and see if they can do anything.
I have exactly the same problem. Mine always showing 100% charge, but as soon I unplug it it turns off and wouldn't turn back on, unless I put it on the charger and it always restarts then. Resetting with home and turn on button doesnt work and not waking up either when I plug it into iMac.
So, I took it to a genius appointment to the apple shop (in Robina Aus.) and they said that they never seen this matter and offered me to replace it for $275, as its 2 years old and out of warranty. I didnt go with this offer, as there is life in my iPad and must be a solution for this...
Let me know if you find a solution