How can I remove a headphone jack that broke off?
Headphone jack broke off inside the device
Your device should still be in warranty, even if it's a first gen Nexus... they came out in 2012, and 2013 has yet to end.
However your kind of accident has happened lots of times: you may find lots of similar questions with several hints. It happened on many devices, but there are common hints you may use. Also, remembering you'd have to be extra careful with an expensive device, you can try with a pair of thin needle nose pliers, trying carefully to grab the jack.
Failing the needle nose pliers, you could try some of this other workarounds:
Straw worked like a charm! Thank you!
you can add a drop of super glue to the end that has the ear pieces insert it into the part where the broken piece is let it dry then unplug it normally.