iPod 3G Won't Charge?

Along with the broken computer I recieved from my friend, I got an iPod Touch Gen 4 and an iPod 3G, I think it's called. The iPod Touch charged perfectly on my computer but the iPod 3G wouldn't charge at all. It would just show the dead battery sign and shut off. So, I thought it might need a new cord cause the old one was pretty beat up so I got this iStuff 3 in 1 Charger from this little shop a few minutes from my house. Once again, it would only charge my iTouch not the G3. When I tried to plug in the G3 to my computer it gave me a little warning saying it didn't recognize it or something. I have a two year old eMachines Windows XP and a currently stuck in safe mode, I think iMac G4. The charger I got looks like thisand it has little attachable things so I can charge it in a wall outlet, car, and USB. Thanks for any help!

A third generation iPod is going to be quite old these days (at least 7 years or more). It's likely that the battery is simply toast.