It's not shutting off by itself, but can this be fixed?

I'm worried because now it seems to be shutting off by itself after a number of hours (still plugged in of course). Can someone help?

My T20 was left on while unplugged and completely discharged. After it was plugged back in, it would not run on battery. The power options says the battery is 99% charged, but unplugging it will turn off the laptop. Can this be fixed?