Turn on iPad speakers with piece of headphones stuck in jack
Afaik that is not possible.

My hint to remove the piece:
Use a stick (or something in the shape of it) slightly thinner than the headphone jack. Than apply superglue to the front. Now go into the headphone jack so the adhesive holds on to the item that is stuck. Now wait a few seconds or minutes, depending on how long the glue takes to really stick. Now you might be able to pull it out.
Be careful with the amount of glue you use, you don't want to get any of that left in the headphone jack.