Macbook late 2008 unibody won't turn on

I recently found my late 2008 MacBook unibody battery was swollen. I removed the battery and have been using it plugged in.

As soon as I removed the battery the MacBook continually goes in to sleep mode. It would turn on, the grey screen with apple logo and progress bar was displayed, then the login screen would show momentarily and then the display and keyboard backlights would turn off, the fan would stop, and the power light would pulse as though it was in sleep mode or the lid was closed. For a little over a week I've been able to eventually get the screen to stay on by power cycling, open/closing the lid, plugging in a MiniDisplay adapter, tapping the spacebar and track pad. I have no idea which of these "strategies" made the difference.

Today however I was unable to get the screen to stay on, and now the MacBook wont turn on at all. Twice the power light turned on momentarily when I pushed the power button, but now nothing happens at all.

Any advice would be appreciated. I've been able to limp this old notebook along for the last 5 years, I'm hoping this isn't the end.