Bless you for having the courage and conscience to do the right thing. Your animals are very lucky to have a steward who takes thoughtful responsibility for them, even when it isn't easy to do so. If all animals had the same, it'd be a whole different world.
Concidentally (or maybe not), today I was listening to an interview on NPR with (the ridculously intellingent and impressive) Dr. Temple Grandin, in which she was talking about the livestock handling systems she's created to minimize stress on cattle as they're going to slaughter. She reminded the listeners that that animals don't have the anticipation of death that humans do, and that they really only know fear and stress, or calm and security, in the slaughtering process.
I agree with the poster who suggested that the easiest and cleanest way are the pruning loppers or even a hatchet, but if you're not comfortable with that, the broom handle will work as well. Any method that doesn't stress the bird, and is quick and clean will be humane.
Best wishes to you, as you do this very right, but very difficult thing.