To kill or operate on your own...that is a tough one...one that I had to make, myself, 2 years ago. My toy dachshund dog was happily ripping my girls apart when we found him. 2 of my hens were so ripped up. I didn't think they could live, but I couldn't bring myself to kill them, so I got out my needle and thread. It took a long grueling 4 hours to get the job done. They did not struggle, or seem to feel pain, or even want to get away from me. They could have been in shock...I don't know. I do know that if what I was doing seemed to be causing them pain, I would have had a much harder time. One of the hens, "Goldy" had been "skinned" down to the bone, just behind her comb, all the way down to her wings. I applied triple antibiotic ointment, then pulled and stitched her skin back together and closed it up all the way. When I got thru with her, I put her in a pet taxi...she immediately started eating and drinking! 2 days later Goldy was laying eggs again!!
The other hen, "Pouffy" was much worse, in that most of one side of her breast was missing...shredded. On close observation I could see a hole had been torn in her croup. I stitched it closed, then put antibiotic ointment all over the breast tissue, pulled together and stitched the wound closed as best as I could. I really didn't think Pouffy could live.
I put water soluable antibiotic and added a vitamin/electrolyte supplement to their water and applied antibiotic ointment to the wounds at least once a day. The smell was terrible for a month! It smelled like something dead, but the girls did not seem to be in any pain. They both ate and drank just fine. I did keep them separate from the rest of the flock. After a month, Pouffy started laying eggs also. She was just coming to the age of egg production, at the time of the injury.
Today, I am really glad I did try to save their lives. They have been good layers always and we are closer friends because of what we went thru together.
All of that said...I do know that the bone sticking out will eventually draw infection into the body. If you can get the skin to come down over the bone, you will have a chance of saving the life of your chicken. As to the quality of life, unless an infection makes her sick, she will be fine with just one leg. We watched a one leggesd wild turkey hen for years. She didn't get too far away from the food and water supply, but she hatched a nest of chicks for several years.
By the way, Frisky, my toy dachshund, was re-homed to my father's house the next day, where she chases wild rabbits all day long. I miss her, but I've heard it is absolutely impossible to break this type of dog from killing chickens, once they start.
May God bless you with wisdom in this discisionl.