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It sounds like your iPod Nano 6th gen will not power on. I found this troubleshooting for you to help resolve the issue from this article:
iPod nano (6th generation): Hardware troubleshooting
- Connect iPod to power for at least ten minutes to ensure that the battery has enough charge to allow iPod to turn on.
- If iPod does not turn on after ten minutes, try resetting the device while it is still connected to power. Reset the iPod by holding down the Sleep/Wake button and the Volume Down button for at least eight seconds or until you see the Apple logo appear.
- If the Apple logo appears on the display, connect iPod to a computer and verify that it appears in iTunes and can play music. If iPod appears in iTunes and can play music, no further troubleshooting is needed.
- If the screen remains dark and will not turn on, then your iPod may need service.