So 2 suggestions
1) Follow iFixit's teardown guide to replace the headphone jack completely
[guide|10826]
http://www.ebay.com/itm/Black-iPod-Nano-7th-Gen-Headphone-Audio-Jack-Power-Volume-Flex-Replacement-/252736334301?hash=item3ad84269dd:g:JTgAAOSwnNBXWcfP
2) I am not sure if this has been tried on a nano, but I've see it used with an iPhone 5C: Use a Lightning port to headphone adapter (the type that comes with an iPhone 7 because it has no headphone jack).