How to cull by neck-breaking. This is my experience and I am sorry to have to share it:
My brother is right handed, so he holds the chicken under his left arm (like a football) pressing its body VERY securely against his side. Then, he puts his right hand on top of the hen's head with the fingers pointing toward the left and the thumb pointing down as if popping a champagne cork, away from you. (I hope this isn't too confusing). Finger knuckles up, so that you have a VERY firm grip on the head and your hand isn't going to slip off mid pull.
Then you want to pull as hard as you can (pardon the graphic nature here, to separate the spinal cord, you will hear the pop and will feel like you are pulling so hard that her head may be in danger of being pulled off, but you're likely not really that strong.
And for good measure, after you have pulled straight forward, bend the neck down (beak to belly, essentially, like folding the bird in half at the neck). Continue to hold the bird securely until any nerve activity has stopped. If necessary, and it likely will not be, but you may have to do it a second time. Our 6 week old rooster gave us a harder time giving up the ghost than our 9 month old rooster, but that is unusual.
If you have heard of the "swing it over your head" method, this is a much more reliable and faster version of that, but the goal is the same- to separate the spinal cord instantly.
The deaths I have witnessed were instant, with only a little shudder, no dramatic bleeding or flailing of wings.
Please know that as awful as it is for you to have to do this, you are doing a kindness to your feathered friend.