lattitude d620 wont run on battery

Just to make sure Windows is reporting battery charge state accurately, press the physical button on the battery itself to have it display its charge level.  It might be on the underside of the battery or you might have to remove the battery from the chassis.  Does that cause all of the LEDs next to the button to light up?  If not, then it would seem your battery isn't charging correctly, in which case it's probably an issue with the charging circuit, and I can't remember on that system whether that's a separate component or integrated into the motherboard.  If the battery shows full and the system still can't draw from it, I've never seen a system that could charge a battery but not use it, but the fix will again probably be a separate part or the entire motherboard.  If it's the latter, given the age of that system, you'd probably be better off (and you might even spend less) buying a newer used system.  You'd probably even get a Windows 7 license out of the deal rather than being stuck on XP or Vista and therefore not getting security updates from Microsoft or any Web browser vendors.