I have one with the exact same problem that I found it seemed to be dead. What I did was unsolder the battery and connected a work bench dc power supply directly to the phone giving it 3.7volts the same as the battery. It turned on so I knew it wasn’t the iPod that was dead. The charger dock connection was bad so I replaced it. I supplied power using the dc power supply and plugged in my charger now it shows charging so the dock connector works. I tried connecting the battery but I think it’s dead because it doesn’t charge or anything. I’m waiting for battery I’ll post when I receive it to see if it finally comes back to life.