What
@neo71665 is saying is that when someone (you) decides to keep chickens you need to be prepared what to do when one is injured or ill beyond repair. Death is part of life and there's no getting around it.
Veterinarians offer euthanasia (put to sleep) for a reason, because it is inhumane to let the animal suffer to death.
If you can't find someone to put the poor chick out of it's misery, schedule an appointment with a veterinarian who will euthanize it. You can probably just drop the chick off there, pay your bill, and leave.