If we wouldn't have found it then it would have died alone in the hay barn on its own, now that its with us I'm at least giving it pain medication. I have no experience with killing chickens, therefore, I will not do it. No one else in my family has the experience either, letting it die a natural death in my opinion (with the help of some pain medication) would be far better than someone in this household awkwardly trying to break the poor things neck - if you don't know what you're doing you can cause the animal to suffer a horrible death, even if it is short.
I had read that if there was a break it would effect the circulation of blood getting to the limb, and the limb would most likely be cold. The chickens injured leg is warm and I can't see any visible break so I do think that it probably is a dislocation - when my mom held the chicken the limb hung down. I wouldn't know the first thing about getting a dislocated limb back in place, I feel like I would just cause it more pain and end up doing more damage.