Q: iPod touch 5th generation won't turn on.

See if you missed anything here:

- iOS: Not responding or does not turn on

https://support.apple.com/en-us/HT201412

- Also try DFU mode after try recovery mode

How to put iPod touch / iPhone into DFU mode « Karthik's scribblings

https://geekindisguise.wordpress.com/2009/07/16/how-to-put-ipod-touch-iphone-int o-dfu-mode/

- If not successful and you can't fully turn the iOS device fully off, let the battery fully drain. After charging for an least an hour try the above again.

- Try another cable

- Try on another computer

- If still not successful that usually indicates a hardware problem and an appointment at the Genius Bar of an Apple store is in order.

how do you make a genius bar appointment? | Apple Support Communities

Apple Retail Store - Genius Bar

https://www.apple.com/retail/geniusbar/

Apple will exchange your iPod for a refurbished one for $99 for the 16 GB 5G and $149 for the other 5Gs. If only the battery the exchange price is $79. They do not fix yours.

iPod Service Center- FAQ

https://support.apple.com/ipod/repair/service