You have your iPod set to Manually manage music and videos (instead of using automatic syncing). If you have not already, click the Sync button (on the iPod's settings screen in iTunes), to see if that action is needed to sync your workout data. If that existing workout data is old (from before the previous time you connected the iPod to iTunes), you may need new workout data. Also, on the iPod's Summary settings screen, there's a button for Reset Warnings. See if clicking it makes the Nike + syncing message/option appear the next time you have new workout data to sync.
I don't recall needing to set or change anything to start sending workout data to the Nike + web site. iTunes prompted me by asking the first time I had workout data on my iPod nano (after my first run using iPod), and I just set it up (including creating a Nike + user account). The user guide
https://manuals.info.apple.com/MANUALS/1000/MA1624/en_US/iPod_nano_2015_user_gui de.pdf (on page 38)
sounds like what I did.
If other solutions do not help, you may need to try a Restore on the iPod. This erases the iPod, reinstalls its software, and sets it to default settings.
Use iTunes on your Mac or PC to restore your iPhone, iPad, or iPod to factory settings - Apple Support
Your existing workout data (on iPod) will be erased. After the Restore, iTunes treats the iPod like a new iPod, so any existing settings related to Nike + on the iPod and in your iTunes library should be reset to default.