Battery Icon/iBook won't restart

I had my iBook G4 plugged in to an adapter to charge the battery when I noticed that the screen kept dimming and then relighting again. It turns out my laptop was going back and forth between battery and adapter for power. I thought it might be the adapter, so I tried another one....same issue. This makes me think it's a battery issue

Now, the iBook battery has died and isn't charging at all when plugged in. I also can't get the iBook to restart at all.

I'd appreciate any help anyone can offer!