Q: why won't my relatively new iPod touch turn on, even after I've charged it for a whole day?

Call your local Apple Store and make an appointment to have them look at it and tell you what the problem is with it.

Try:

- 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/