When you have an animal in your care that is suffering from illness or injury, it is depending on you to provide the final kindness by giving it a humane end.  Humane = fast & effective.  It's not time to worry about your feelings, but instead pull up your britches and do what's best for your animal.  
I know it's tough to dispatch a baby chick, but really, the fastest & most effective way is to snip through the neck with a pair of sharp scissors.  Poultry shears or kitchen scissors can do the job well.  I don't like to watch it happen, so I do it with my hands inside a paper bag or at night over the hole I've dug at the drip-line of a bush.  I can hold the chick with one hand and position the scissors open at the neck.  Then without looking I can make the quick snip and let the pieces fall into the bag or hole.  It's effective because once the neck is severed the chick has Crossed The Road.  Although there may still be some reflexive movement from the chick's body.
When I bury the chicks at the drip-line of a bush I know it will then contribute to the growth of the plant, and in that way, continue on.