That particular Dell notebook (I have one.) has a little "paper-clip hole" on the right side of the case. If you shut down the computer then insert an appropriate tool there (Don't use anything that might break off -- like a pencil or toothpick!) and press in gently you'll reset the BIOS. I don't think it's likely to help, but it's possible, and it's certainly worth the attempt.
The problem you're describing is probably not due simply to failure of an old battery but, rather, to charging subsystem failure or misconfiguration. If you're lucky the BIOS reset will fix it, if you're slightly less lucky a battery and recharger replacement will fix it, and if you're not lucky at all then a MB repair / replacement will be required. That's an old system, probably not under warranty still, right? Mine just went out of its three-year warranty last month. It has suffered two motherboard, two hard drive, one screen and two hinge failures. Not one of Dell's better efforts. My experience with more recent models is scarcely better. I've had it with their portable computers. Most of the failures, I'm convinced, are caused by the cheap, flimsy chassis / shell materials they use. The flexing they undergo during normal use causes connectors and electronics to go through hell, hence the high failure rate. I'll never again buy a notebook in which I can feel any flex whatsoever when I pick it up.
I'd try the BIOS trick then, if that doesn't work, talk with Dell about it.
Good luck!
- prosaic
Edit: Oops! While cataloguing the failures of my Inspiron 7500 I forgot to mention the video card failure. I had the next-business-day service agreement. Not one of these failures took less than a week to resolve! I think that, if I ever do business with Dell again, it will be for desktop systems or, possibly, servers -- but not for portables.