Not to be rude or mean or anything, but I agree with
@IamRainey. This method does not sound very quick and easy to me. I'm very sorry about your hen.

If you can't handle using an ax, just have someone else do it for you. Think about it this way: It's what is best for the chicken, not you.
Using the dry ice method might be easy and good for you, but for the chicken it is stressful, terrifying, and then she can't breath.
Using an ax might be stressful for you, but the chicken will be dead in less than a second and won't feel stress and pain.
You can use the dry ice you want, she's your chicken. But I just wanted to let you know about this. You might rest easier after she is gone, knowing she had a stress-free death.