This is what I see in her eye:
View attachment 1882328
Her eye is in the back, should I try poking the swollen part in front with a needle?
I don't think I would poke it with a needle. Chicken pus is a semi-hard cheese-like substance. You may be able to put warm compresses on the eye to help loosen it a bit, then press it out. Having a helper is a very good idea.
Do you have any antibiotics on hand?
Any foul odor coming from the face?
https://www.backyardchickens.com/threads/maran-pullet-with-swollen-closed-eye.1284967/#post-20846044