A big AWWWWW for cute lil' baby chicks!
And a big
for having to cull one, but
to you for taking the responsibility for it. This is something every chicken keeper needs to consider and have a plan in mind for how they would quickly & humanely put a chick or chicken out of its misery. Sometimes you could be faced with a bird who is injured so badly and would need help to Cross The Road right away.
I think the scissor method is the fastest, therefore the most humane, but it can be difficult for a person to do. But I'm going to describe a way that makes it more bearable. Take the chick and wrap it in a paper towel, talking soothingly to it the whole while. Hold the wrapped chick over an opened paper bag, cardboard box, or a hole I've dug at the drip-line of a tree or bush. Locate where the neck is under the paper towel. (You just cover it with one layer of toweling, don't wrap it like a mummy) You can use any kind of sharp scissors or shears, they can be kitchen scissors, poultry shears, big office scissors, etc. Just about anything except your kids' little plastic craft scissors. Use the back of the scissors to cut so you get the best leverage. You should be able to snip completely through with one cut. Then let both pieces of the chick, the head & the body, fall into the bag/box/hole. There will be some reflexive movement from the body but the chick is beyond feeling any pain, once its head separates from the body it has crossed to The Other Side. Keep an extra paper towel handy to wipe the spots of blood off the scissors, you can do this without looking if it bothers you.
It really takes longer to read about it than to do it. It's quick & effective and therefore the most humane method I've found. I just don't like holding them or seeing them afterwards, that's why I use the paper towel & the instant disposal.
Now let's hope we don't have to do this for a long long long long time!