I've started having this trouble just after having updated to the new iTunes, about a week ago now.
After some research, I've discovered that Apple stopped supporting the iPod nano (July 27, 2017) at just the same time I stopped seeing the Nike plus app on iTunes, they just turned the function off I guess. I suppose this way, Apple doesn't have to provide support for the nano any more (I really doubt this is some sort of bug, even though I hope it is).
I love my nano for running, its fits so nicely into my outfit and general routine.
But if I can't use it track my runs anymore, then Nike and Apple don't have the sort of fitness tracking solution I'm looking for (I don't want to run with a big phone in my pocket, and I don't like to wear a watch when I workout either -- I suppose my kids just inherited an iPod as well.. bummer đ).