Too close to the end

Today while I was working on the coop I noticed Big Red standing at the corner of the chicken run.  She looked a little funny, but I didn't think much of it since the others hurried over to the corner with her, wanting to be let out.  

About 30 minutes later as I walked by, I saw her again and knew something was wrong.  I opened the door and while all of the birds reacted normally, hurrying over to get out and free-range, Big Red sort of turned half an inch but didn't move.  

I scooped her up and immediately knew there was something wrong and it was probably that she's egg bound.  None of the birds stand around like this and none of them would let me scoop them up the way she did.  And, as soon as I had my hands around her, I felt how huge and swollen her abdomen was.

Being egg bound means that an egg is stuck inside her.  If she can't pass it within 24-48 hours, she will die.  I don't know how long she's been like this, so I'm not sure how much time I have.

I carried her into the house and with one hand cleaned out half of the sink and filled it with warm water.

Never did I think I would have a chicken in my sink.

Sorry for the dirty lady in the picture, I didn't think I'd be posing for any pictures after all the yard work.

After a 30 minute soak I put her in my shower so she would be contained and away from any excitement outside.  She immediately went to the bathroom, but there was no evidence of egg.  I gave her food and water and saw that she age, but she wouldn't drink.

A few hours later, I made a tub of warm water with Epsons salt and sat down with her.

You can see her eyes closing in this one

She soaked for about 25 minutes while trying to squeeze something out.  I can tell she's uncomfortable, but I'm just not sure what else to do.  Even my internet research isn't helpful as they say I should be able to feel the egg inside her, which I can't, and some of their solutions involve going into her vent, which I know I can't do.

I bundled her up, dried her off, and then put her back in the shower with a heat lamp, water, and food.  I'm going to take some yogurt and molasses to her in a few minutes as that's what another chicken friend told me to do.  

I don't want her to suffer but after reading up on how to "humanely" put her down, I'm sure I can't do any of those things either.