Your hen is suffering and the vegan neighbor would surely want you to end it. No sense in wasting time making phone calls or waiting around for an appointment. We had a nasty nasty rooster, he got the shotgun treatment. I figure if we ever have a terminal hen she'll get the chopping block. It's probably a 2 person job - one to stretch her neck out and the other to swing the axe. Call a supportive friend to help you. Don't do what we did years ago with our very first flock - we had a sick hen and we just let her keep hobbling around until she finally collapsed and died. That still bothers me and I wish we had put her down rather than let her suffer. After that I don't know what to do with their bodies? We lived on 4 acres so we burried them. But now we only have a half an acre and don't want to start a graveyard since we are on a well...