Headphone jack broke in iPad

First, completely power down your iPad.

You must be extremely careful, but you will have to use a drill bit fine enough and drill out the piece stuck in the port. Go VERY slowly and don't force the drill in. You want to avoid drilling through the bottom of the headphone port. Once the piece in the port is drilled up, it should fall out.

Just make sure the drill's diameter is dramatically smaller than the port, avoid hitting the insides of the port, and don't push through the bottom.

When you're done drilling, the dust and loose metal bits should fall out. Make sure you blow all the metal particles out from the inside of the port.

You ought to wear safety goggles, too. Getting metal particles in your eyes is not a good thing.

I've done drill-out repairs like this before. If you're careful and steady with your hands, it should only take a few minutes. :)