My Aunt had a California Grey chicken 2.5 years old with this problem not too long ago too. We put her warm water bath, gently massaged the area to help ease it out. It was a total of 4 baths over 5hrs. The egg broke part way out and was the foulest smelling thing. The membrane was holding the shell together and after the contents came out of the egg the shell was able to be removed, she actually pushed it out. About a 1/2hour later she passed another egg that was normal but also jumbo in size.
We put her on antibiotics (don't remember what one), put ointment on her vent after gently rinsing out her vent to check for infection, and finger probe to check for other egg or if anything else could be felt wrong. She was given some probiotics, after the 5 days of antibiotics, feed was a mix of wet pellets (mushy) and oats, was given ACV in her water and was kept seperated in a cage in the coop fairly dark so she wouldn't try to produce another egg.
She is the "Ole Grand Dame" of my Aunt's flock of 24 mixed birds. She is doing fine and didn't have any further problems. She is checked every day now to make sure that it won't happen again. It amazes me the things that chickens can recover from and still be happy. My Aunt knows she is getting past her productive time but as she is the last of her first flock she will keep her until the time comes to put her to sleep or she leaves on her own.
**I forgot to mention olive oil on the egg can help for it to come out too.**
