Vent prolapse/culling

My sweet Americana, Nugget, has a vent prolapse that I’ve been managing at home as best I can but it’s gotten bad. It happens every time she lays, many times it bleeds, and she is often separated from the rest of the flock for rest. I have a sinking feeling I’m going to need to cull her soon.

I’ve never had to cull one of my girls, and I’m nervous about this and want to do it as painlessly as possible for her when it’s time. What is the best way to cull that would be fast and painless for her?