Hugs to you.

We don' know each other, but it pained me to hear that you seem to think you can't help anymore, that that is what you feel you've learned from your chicken breaking her leg. I think you probably feel the helplessness that we all feel when there is nothing more to be done for our animals (and our people loved ones too). It is a helplessness that defies logic, and our brains turn to guilt, we second-guess, we go over everything again and again, we subconsciously feel there must be a reason we feel so bad, that we did something bad, or did it badly. But it didn't appear that way to me.
When we can usually do so much for our companion animals, it is naturally very hard to accept when we cannot do more than help them pass on and not prolong their suffering. It takes lots of time to process that helplessness emotionally. Sometimes however the time to act is pressing, because the animal is suffering all the while. So often we must act before we are "ready". That is pretty traumatic, on top of the distress of the original accident.
But I think you were quite heroic in helping her. You did not fail her. You quickly gathered information and help from your friends and knowledgeable people, and carefully weighed the options, and made the decision you pretty much had to make. All within the same day, as I recall. It was a very big deal. Don't listen to anyone who tries to minimize the experience. Your pain of loss is real, the trauma of the event is real, and hopefully you will adjust to it over time.
You are likely experiencing some PTSD actually, most people do IMO. The guilt and nightmares are you processing the whole experience. Be kind to yourself. Look into things that help with PTSD and try them. I'll say again: know that you did not fail her, quite the opposite.
Let your pain be a reflection of the love you feel for her. The honor her, and let that love shine on as you continue to care for the ones still here.