There is a humane way to do it.... search "humane chick culling" or something like that. I think one way was to put some gasoline in a box, close it with the chick in it, and then the chick will fall asleep then die.
Sorry to hear you have a sick one. A box with gasoline is NOT humane. Scissors or garden shears is. Easier than pulling it's neck if you need to cull it. As far as I am concerned, if you aren't capable/prepared to cull, don't breed. Nobody hacthes birds to kill them as cute fluffy chicks, but sometimes due to unforseen problems it is necessary.