get yourself a
Dumont Nr. 5 or visit your local anatomy, neurobiology, surgery lab/department etc.… or electronic store, they probably will you lend one for a few minutes when ask nicely. Depending on what kind of glue you used (did this really ever worked?) you will be able to clean out the port.
Right. Poke a metal forceps into an electrical connector (or for that matter, a metal screwdriver, metal screw, etc.). Let's hear it for potential (pun intended) short circuits and electrostatic discharges, and for harder metals damaging the softer metals that make up the port's electrical contacts.
Whether to remove a broken-off connector or to fish-out accumulated pocket lint, a softer, non-conductive tool (wood or plastic) is a safer way to go. It's true, plastic and wood are less likely to be successful when brute strength may be required, but they call it brute for a reason - what good would an empty port do, if the port is permanently damaged by the "fix?"
This is a fairly common repair job for third party repair shops to replace the Lightning flex cable and port. Should cost less than $50 for the job.
Would a shop in the First World charge as little as $50 to do that job? Perhaps for an iPhone, which is much easier to open and re-seal. iPads are significantly more trouble to service in the field. I'm seeing far higher rates for this job for recent-model iPads (like the iPad 8) - half the price of a new iPad, or even more.
If you have an Apple Store nearby, bring it in for an evaluation. If they can fish out the broken connector at no charge then it's a "win." If not, you can decline the repair quote and decide whether you want to shop around for a cheaper repair.