First, I know how difficult it is to make this decision when you've nursed a little one along and tried so hard to make it well. I can tell you what I did, for what it's worth, but I know others will disagree with my method.
The last of 17 chicks to hatch under my hen was very slow getting out of the egg and eventually had to have the shell peeled off of him. It became obvious within a few hours that one leg wasn't working right, and he was never able to support himself or walk. We discovered that the hip on that side was congenitally dislocated, so my daughter fashioned a brace and corset to hold it in place, hoping his malleable bones would eventually grow correctly. We hand-fed and babied him for about 3 weeks, but his condition got worse as he began to grow. I couldn't watch him struggle anymore.
My chick was still tiny, and after reading about various methods I simply wrapped him in a soft cloth and held his body in one hand while pulling his head straight out with the other. The neck breaks and death is instantaneous, though the body may spasm for a few seconds afterwards as the nerves react. The instructions I read said to pull the head straight out with steady traction rather than a jerking, quick motion. As I said, my chick was tiny, and I unintentionally pulled his head off.
I've had to have a lot of pets euthanized over the years, and I've come to some personal conclusions about which methods are most humane. I probably would not have a vet put my horse, goat, duck or pet rat "to sleep" again in the future because of the length of the procedure and the anxiety involved. I would caution you against using carbon monoxide poisoning because it's painful and traumatizing, no matter what you may hear. IMHO the best way for the animal is the quickest, even though it may not be the pleasantest way for us. As caretakers it's our responsibility to put their needs ahead of our own discomfort and squeamishness.
I'm sorry..