You're speaking with Dr. Michael Salkin. Welcome to JustAnswer. I'm currently typing up my reply.
Let's get rid of this feather. Sky will appreciate it even if it hurts for just a second. It's best if you have help. She'll need to be carefully restrained - never compress her body because she needs to breathe with her whole body (she doesn't have a diaphragm) - surround her with your hand +/- light towel and follow the shaft of that broken feather to where you see it leave her skin. Grasp the shaft near to the skin with a grasping tool (pliers or similar) and quickly pull it out of the skin. You may see a drop of blood where it comes out but bleeding will stop immediately. And that's all there's to it.
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