Big props for taking the time to plan everything so your hen could exit quickly. It sounds like she was very woozy already, so she probably simply stopped feeling pain when you did the deed.Sounds like it was harder for you than the bird. So that's a good thing! You did your best for your bird and I am sure she appreciates you.
This experience is how everything I do with chickens seems to turn out: plan the least stressful path for the birds; run into a series of figurative walls as I assemble all the necessary tools, usually in awful weather; and finally, end up spending as much time cleaning up as preparing. But in my story, there would be at least one chicken on my head with squish poop on her feet.
Thank you for sharing. Virtual hugs.