This is an essential question I think that every chicken owner must consider, and should know their own answer to it long before it ever becomes necessary to act on it. The longer you keep chickens, and the more of them you have, the more likely it will be that one day, some day, you will be faced with a bird that is suffering from illness or injury. That bird will be depending upon YOU to put an end to its misery and help it get Across The Road. There may not be time to go to the vet, maybe even not time to wait for a friend or relative to come help.
There have been many threads written on just this topic, do a search on words like "Humane way to cull". You'll see that often the methods that are the most difficult for the person are the kindest to the animal, because they provide quick & efficent results. You'll just have to strap on your Big Boy/Girl boots and do it.