Hi,
Unless the cable/Usb port or charge port is faulty and you have 3u tools installed on your computer you can try starting the app then connecting the iPod with the lightning cable.
“When an Apple device is in DFU mode the screen is completely black with no sign of life”.
If you are able to see device detected in DFU mode on 3u tools, you can proceed to flashing the device with the firmware you have or download the one you need through 3u tools firmware options.
If not then follow the procedure for getting the iPod into DFU mode (different to restore) if you see the connect to iTunes logo, you are not in DFU mode.
If you can’t establish communication with 3u tools and the device then there is a fault at either the USB port/Lightning cable/Charge port/Flex cable or Mainboard.
Hope this helps