Redhen, I'm sorry for your loss but am glad you & Mister took care of your hen. Whether you do it by yourself in your arms, or use the broomstick method, or have someone help, cervical dislocation is a quick & humane method of culling a chicken. A firm, swift, decisive yank kills them instantly. Once you feel that crunch and see the neck hanging limp, you know the job has been done.
Tothemoon, you can use this method with chicks too, hold their little feet in one hand, place their neck between two fingers, & pull. But it's too heartbreaking for me to see so I instead wrap them in a paper towel, hold them over the little hole I've dug under a bush, and snip through their neck with a sharp pair of shears, and let everything fall into the hole. Well, not the shears. It's up to you to decide just when their quality of life is so poor that it's best to help it out.
As for dispatching meat birds, if you don't have a cone you can just use duct tape wrapped a few times around their wings & sides. And zip-ties to hold their legs together and hang them overhead.