More than likely the battery is dead. Putting a battery through charge/discharge cycles slowly damages it. The damage shows up in terms of how much current the battery is able to put out. The more damage, the less current it can output.
During the early stages of battery failure, the reduced current output shows up as lower voltage. The more current you pull from a battery, the lower its voltage drops. This shows up in the battery % as the percent remaining charge dropping faster than it did when new. And so your old battery doesn't last as long as it did when it was new.
During the late stages of battery failure, the current it can put out drops below what the device needs to stay powered on. So the battery will appear to power the device from 100% to (say) 40%. Then suddenly the device will turn off.. What's happened is that from 100% to 40%, the battery could put out enough current to keep the device powered on. But after 40% it couldn't put out enough current, so the device powered off.
In the final stage of battery failure, the battery cannot put out enough current to keep your device powered on even when it's at 100% charge. So it will do exactly what you're seeing - show 100% charge, but turn off the moment you remove AC power.