I totally understand if you don't feel up to some of the more violent or 'hands on' methods of putting your hen out of its misery.
I plan to have meat birds at some point, and I think I will be okay with processing, but I cannot imagine having to kill any of the ones that I keep as layers. They are all my pets and I am very attached!
That said, I wouldn't want them to suffer and die slowly if something like this were to happen to any of them, so I would force myself to do it for their sake.
One option you might consider, that might be less traumatic for you, and still humane for the hen, is to use engine starter fluid (you can get it at auto supply stores). We were researching ways to get rid of a pest sparrow colony that was attacking our purple martins, and wanted info on what to do with the ones we trapped. The instructions we found for humane disposal involved placing the birds in a small, airtight space (like a bucket or a plastic storage container). Then you soak a sponge or rag with the starter fluid and place it inside with the birds.
Apparently the starter fluid stuff's main ingredient is ether, which is a powerful anesthetic. The birds quietly go to sleep and don't wake up.
ANY way you choose to do this won't be fun, but at least you know you hen went in a way that was painless and is no longer suffering.
ETA: I am very sorry that this has happened to your sweet little hen, I know how attached I am to all of mine, so I know how hard it must be to make this kind of choice. It sounds like you've been giving her excellent care, though, so she has had a great home and a great life.
