iPod Touch Will Not Sync or Charge

I have an iPod Touch, and when I connect it to my computer, it will not detect the computer (so it won't charge) and the computer doesn't detect it (so it won't sync).

I can't restore, I can't take it to Apple because it's jailbroken, and I can't get the spinning wheel of death because the battery is dead.

What should I do? I took the cable to Apple, and they tried it on one of their iPods, it worked, so it's a problem with the iPod. It's not a problem with the USB port either, because I've tried other USB devices, and they worked.