A pair of loppers or an ax to separate the head from the body is quick. I usually cut the jugular vein to let them bleed out, and many use the broomstick method of cervical dislocation. The latter may take a little practice. Do you have a hunter friend who is used to putting down an animal?
Sorry about your hen. Her injuries are too bad to help her. Her intestines are exposed, and she will probably die a painful death from infection. I would recommend putting her down.