Thought I would chime in here, as I've also just had to put down a favorite hen who was very sick.
There are so many methods of "culling", and everyone will favor what they're most comfortable with. Last year I spoke with a local chicken breeder and Judge about some health issues I was having with a couple birds. He had several suggestions, and the one I recall the clearest was "Sometimes the best thing in your medicine chest is a good, sharp hatchet". As my birds are mainly pets, it has taken me time to come to terms with this. I've lost birds for different reasons, mainly just passing on their own accord- some suddenly, and some after a week or more of illness.
I have always felt strongly that part of owning animals is making the best decisions on their behalf, and not allowing any to suffer needlessly. After trying everything in my power to heal sick birds, and she's just going downhill, it's "time". However, as someone who has not grown up with raising meat animals or hunting, I have struggled with this. And, I don't feel right about bringing the brother-in-law or my husband to do my "dirty work". After reading all of the different methods of doing in a chicken, I have found that the hatchet method does indeed work the best for me.
I have a stump that I have 2, 3" screws about an inch apart, screwed partially in on one side of the top of the stump. I go grab my heavy butcher knife, make sure it's SHARP (did this with a rather dull axe a while back, was not a pretty sight.... Sharp, people. Sharp.) and I put that by the stump with a box or whatever I'm going to put the bird in afterwards. I go and get the bird and wrap her body snugly in a hand towel to contain the post-chop twitching and reduce any struggle stress prior. I'm pretty sentimental and it takes me some time to work up to do this even though I know it's the kindest thing to do for the bird. Actually, I find it really upsetting. Especially with the birds I've grown fond of. When I've said my goodbyes, I lay the bird on the stump with her neck between the screws, gently pull her body so that the neck is straight-ish and give a very quick, hard chop. There is very little twitching, and I've found it a bit easier to hold onto the body with the towel than with just one hand. I prefer to do this by myself so this method works for me. So for those of you who are also working up to ending suffering of your favorite birds, know that the lead up to the chop is the WORST part. The chop itself is so fast, and I feel it's the least amount of trauma and suffering. There is honestly not that much blood, the bird generally closes their eyes right away and it's over. No more suffering for your bird, and no more stressing about it for you. Every time I have had to do this, I still have to remind myself of all these things. Someday it will get easier, or maybe not. Anyway, I just thought I'd share my take on it.