How can I remove a headphone jack that broke off?

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:

1. Use a straw, the kind you'd use to sip a drink. Plastic, soft and inexpensive.: cut it so you can "roll" in the socket, then push all the way into and try to move it around. If you're lucky, it should "disengage" the broken tip and the spring keeping the connector pushed on it, so, pushing carefully the straw out, the plug will follow
2. Failing that, but having to apply the utmost care to avoid further damage, you could try applying some fast-drying but strong glue on a Q-tip without the cotton or other thin piece of plastic and/or metal. Eventually using the straw to avoid smearing glue and damaging everything else, connect with the broken jack, let it dry and gently push out