Awww, I'm sorry for your situation.
Like others have said, this is a sad but necessary aspect of keeping chickens. Sometimes it must be done for the benefit of the suffering individual as well as the well-being of the rest of the flock.
I dig a deep hole at the drip line of a tree or bush in my garden. Then I hold the bird under one arm, get it calm & relaxed, bring it to the garden, then give the head a decisive yank which breaks the neck and dispatches them instantly. Their wings will reflexively flap for a few moments afterwards, I continue to hold them until they're still. Or place them under a bucket & walk away. I bury them near a plant so their bodies will help it to grow and in that way, they continue to live.
I advise all chicken keepers to consider this issue right from the start, along with other decisions about managing their flock. Keeping chickens is different from keeping other types of animals, often we have to learn to care for many of their needs ourselves. Professional care for them is often unavailable, or too costly.
Sometimes it helps to have a more impartial person do the culling for you, they can focus on doing the job quickly & humanely and not be as emotionally attatched to the individual animal. You can even trade this "favor" with another chicken keeper nearby.
It's a very sad task, but sometimes your chickens are depending on you to end their suffering quickly and humanely, or to keep them safe from another bird's disease. If they were out in the wild their impairments would probably make them vulnerable to predators, who would use less kind tactics to dispatch them. I wish you courage & fortitude to do what your hen needs you to do.
