I bought another Nano. It wouldn't work either, but after further searching, I found an answer that worked for me:
"Open the Windows File Explorer on your Windows 10, Windows 8, Windows 7, Vista, Windows XP. Browse to C:\Program Files\Common Files\Apple\Mobile Device Support\Drivers. You should then find four files from this folder. There will be a file named “usbaapl.inf” on 32 bit system or “usbaapl64.inf” for a 64-bit version of Windows. Right click on this INF file and choose Install from the pop-up menu to quickly install the Apple Mobile Device USB driver again. If there’s no Drivers folder, look in C:\Program Files (x86)\Common Files\Apple\Mobile Device Support\Drivers.
When you have enabled, uninstalled & re-installed, installed the Apple Mobile Device USB Driver on Windows PC, remove the iPhone iPad or iPod from USB, restart the computer. When you log in Windows system again, run iTunes first, then connect up iOS device (iPhone, iPad or iPod) to computer USB port to verify that the device is recognized properly. Make sure to load iTunes program on PC before you attach your iOS device to USB."
http://www.softwarert.com/iphone-ipad-ipod-detected-not-recognized-itunes/