As an alternative to repair and improvised extraction (which btw are the best answers):
Bluetooth to 3.5mm Rechargeable Audio Adapters are affordable, and very real. Combined with a 3.5mm coupler, you'd have Bluetooth on-a-stick for your headphones and iPad 2. Best answer? Nope. Coolest Answer? Just maybe. :)
Excelvan BT03 Bluetooth is the one I use from time to time, in those events where I need to isolate hardware from software in iProducts and Macs. You'll need a 3.5mm Female-Female coupler to get your headphones into it.