How do I fix an unresponsive Apple TV 2nd Generation?

Try this (these instructions may be slightly outdated):

Get yourself a micro USB cable that supports data transfer, you can restore your Apple TV from iTunes:

1. Remove ALL cables from Apple TV. (if you don't you will not see Apple TV in the iTunes Source List)
2. Connect the micro USB cable to the Apple TV and your computer.
3. Reconnect the power cable (on some models, this is not necessary)
4. Open iTunes.
5. Select your Apple TV in the Devices list (which can be found below the volume slider near the top of the iTunes window), and then click Restore.

If your Apple TV fails to restore through iTunes……..

1. Check that you have correctly followed the instructions.
2. If not already, try using a powered USB port.
3. Repeat the process but this time try connecting the cable while iTunes is closed.
4. Repeat the process with another cable.
5. Try restoring the Apple TV on another computer in another location.

That model is officially obsolete and practically useless. Replace with new.

Not necessarily, older model Apple TVs still work great as AirPlay receivers for things such as a stereo system or a conference/classroom display via Screen Mirroring. You can get them for rather cheap secondhand, and they have all the ports necessary for most use cases (digital audio, ethernet, micro USB for maintenance).