I have laid mine on their back. They are very calm and close their eyes and relax. Then I place a very very sharp knife blade right under their chin - this doesn't upset them either, and cut quicklly, cutting the artery. The head nearly comes off in one cut. Even the first one I had to do two cuts wasn't upset by my fumbling, she didn't struggle. I don't know why - I was very surprised. I've heard they go into a kind of trance lying on their backs. Then I lay a heavy towel over them while the wings flap in the death reflex.
I don't think they realize what is happening. Because one day for the first time, right before I cut, and she was lying peacefully on her back, I told her to "go to the light." She jerked open her eyes wide, jumped up frantically and took off. I guess she didn't want yet to go to the light. So I don't say that anymore.
Laying a chick down on its tummy and hitting hard with a heavy knife cost me $1500 in emergency room fees. I missed the chicks neck and nearly took off the end of my thumb on the hand holding the chick down. Lots of stitches, lost my thumbnail for awhile.
I'm very sorry for your chick. I have one too three weeks old and gags when she eats and can't eat much. Her size is very small now compared to her sisters. And she farts all the time now. Something wrong with her digestive track I think. I'm watching how its going with her closely.