Forgot to update.
Later that day I was able to grab the hen, which was surprisingly easy. She was so worn down and exhausted. Upon examining her I found that because of her inability to move or take care of herself she was absolutely infested with mites. I've never seen anything like it! All of other hens were completely fine, though (gave them all a DE bath just in case). I laid her back down and went to the garage to sharpen my hatchet. When I came back 30 min later she hadn't moved from the spot I laid her down in.
She was ready to go.
I have never had to put a hen down, so this was my first. I watched a youtube video and made a few practice swings on a branch to ensure a somewhat accurate swing. Her death was quick and painless. One swing. Complete and perfect separation. Sad, but I guess that's how it goes. The rest of the hens are all vaccinated, so I hope they don't suffer a similar fate. Thanks for helping to ease my conscience in this decision. It was the right call.