After this current crisis is over it would be a good idea to do some research and find a way to cull that you can be comfortable with. I'm a big proponent of what is hard on us is usually best for them and think a quick strike of the ax is far more humane then many of the easy for us methods I've seen--putting them in the freezer, a box hooked up the car exhaust, etc.
I did see a thread on using dry ice the other day though that looked promising. The poster put the dry ice cube in the bottom of the box and placed a wire wrack over the top that she then coverd with straw to make a thick bed. The sick pullet was then taken off the roost and placed in the box (the whole room was kept dim so she'd stay sleepy) the lid of the box was sealed and when she checked 20 minutes later the pullet was gone. She said she stayed in the room the whole time and that there was never any struggling or clucking from the pullet. I don't know anything about how the chemical reaction with the dry ice works and will continue to use more direct methods myself, but it may be something for you to look into so you're better prepared next time.
