I know this may be a little late, but I've seen people euthinize chickens and ducks using a little more "hands off" approach.
For little chicks that had gotten mauled by a cat, a friend put them in a paper bag and put it in the freezer. For a larger bird he had put it in a trash can with a lid and ran a hose from the car exaust and into the can. Both time they just went to sleep in less than 5 minutes.
Now I'm not saying that using a knife isn't quicker and less painful, but definately not cleaner or easier. With both freezing and carbon monoxide poisoning they just go to sleep.
I'm sorry you had to do this, but on the other hand it is heartening to see that you were responsible and did the right thing and end the poor little one suffering. Don't get me going on people who neglect animals and their quality of life. Suffering sucks and I don't wish it on anyone or any living thing. Quality of life is always better than quantity.
Peace & Aloha,
Cory