MX6027 battery woes

I'm looking for some help with my wife's old Gateway MX6027, which is having some battery problems.

Here is the backstory: A few months ago, her power cord started to go bad, flickering on and off, not charging the battery etc. We limped it along, but then her battery started not holding very much charge, and the red led on the front (low battery indicator I think) would not turn off. Once the battery lost all its charge, the laptop would shut off if you bumped the cord and momentarily lost connection. Then the computer started losing power even if the power cord was not bumped. We stashed the laptop for a few months until I got around to working on it.

I bought a new power cord and battery for the computer. But this has not solved the problem. When the battery first arrived, I believe it was at least somewhat charged, because I could run the computer without the power cord. I noticed that the battery monitor indicated that the battery had 0% power, but that it was "charged" when the AC was plugged in, not "charging."

I let the battery run down to see if I could charge it. Now if I plug the battery in, even if I am running on wall power, the computer shuts off after a few minutes. In fact, as soon as I plug in the battery, the screen dims and the tray icon switches from the power cord to the battery. It still tells me the battery has no charge. Leaving the battery in and the laptop plugged in while shut down does not charge the battery. The red led stays on, and when I turn it on it sometimes doesn't even make it past the BIOS.

Any ideas what is wrong? Could it be a problem with the motherboard? Charging circuit? Thanks.