Sick Hen - Was Egg Bound, might have an infection? Please help! *UPDATE* still swollen :(

Background - Yesterday evening noticed hen was listless, sleeping in middle of yard. Today still acting listless, was able to catch easily. Vent had a foul smell, swollen, black and crusty.

Treatment so far - Rinsed with clean water, pulled out a broken egg that smelled terrible, looked like it had some pus and blood in the discharge, rinsed vent with clean water and seems like there is no more egg or egg fragments inside. Hen is still acting listless.

Next Step - I'm going to bring her into the garage and isolate her in the sick pen to keep her quiet and in the dark so she'll rest and I can monitor if she is eating and pooping. Due to the appearance of the pus I think she may have an infection and I'm thinking of putting antibiotics in her water. I'm planning to feed her some scrambled eggs and yogurt to help her build up her strength and rebuild her gut from the antibiotics.

Question - Am I missing anything?

ETA: We brought her in last night and she didn't want to sit down or lay down and her belly felt a little swollen and hot. We cut away the feathers that were dirty so she wouldn't continue to get build up. I examined the vent and could not find anything else, lubed it up to hopefully help her pass whatever she's struggling with. When I was doing that a lot of yellowish liquid came out and after that she was able to lay down to sleep. Her comb is still red and she is eating, drinking, and pooping. She's still unusually calm.

Should I try the hot bath?

 

Well she does seem much better today. She's still eating, drinking, pooping. She looked like her belly was still swollen a little so I did a hot bath which she seemed to like. Then after I got her mostly dry I let her peck and scratch in the yard. She was clucking and able to walk and scratch and act "chickeny" which she could not do yesterday. She heard her flockmates in the back yard and wanted to go to them so I let her into the yard for awhile. She's been scratching and running around like normal.

I'm thinking about bringing her back into the garage though to have her drink the antibiotic water just in case and give her eggs/yogurt and let her rest.

Any thoughts?