No matter where you are on the spectrum of chicken keeping, whether you have them as pure nameless livestock or indoor diaper-wearing pet chickens, you must consider YOUR personal answer to the question above. Because it's likely that at some point in your chicken-keeping years you will have to face this issue with at least one of your injured/ailing chickens.
Just as there is a wide spectrum represented among our BYC members in our purposes for keeping chickens, there is a wide range of choices to use for culling a chicken. Some will seem more appealing than others, in terms of cost, personal involvement, and sensitivity. I think it's a good idea to explore all the different options, and then make YOUR own decision for YOUR own chickens.
I don't see where the OP placed themself on the sensitivity spectrum, they seem to be asking a basic question. The cervical dislocation method described IS as humane as any. Some people use a stick held down with both feet, instead of one foot, to hold the head down, does that seem a kinder way to dispatch a feathered friend? It's the method I use with old layers that must be culled, it's easier for me than to give their loyal old necks a chop with an axe.
If you cannot bear culling your own birds by yourself, and cannot afford to have a veterinarian do it (I would love to find a local vet that will "do it for no cost" like so many people claim to have found) perhaps you could find a friend/relative you can trust to do it for you. Or even offer to trade the favor with another local chicken keeper, they cull any of yours and you cull any of theirs. Sometimes it's easier to do the job efficiently, thereby humanely, when you're not as emotionally attatched.
RoosterJerry, I'm sorry for your ailing hen, I hope you find a way to end her suffering that also works well for you.