I'm so sorry this is happening to you. I've gone through this twice with one hen first, then several of them getting severely ill with a coccidiosis outbreak. In both cases, they went down "quickly enough" where I didn't end up having to resort to "doing the deed" myself, but I did do some research.
Some people suggest a method that uses baking soda and vinegar to make a Carbon Dioxide "gas chamber", but one poster on here said that they had done it and that the bird panicked with the smell of the vinegar (or something) and it was NOT PRETTY.
However, I did get a suggested variation on this from a veterinarian I know. It's the "gas chamber" idea but using Carbon MONOXIDE instead. Of course you need to be careful with this as Carbon Monoxide is deadly.
You could rig a large Rubbermaid container with a lid and a large pipe/tube going into it, and tie the tube in with your car exhaust. It will smell a little funny to them, but they will go to SLEEP quickly and in a non-violent manner (shouldn't thrash, etc.). Carbon Monoxide can knock you out at pretty low levels - we had a leak once at my lab and one of my co-workers was just walking down the hall and passed out!!! The rest of us in the office/lab had headaches and were just plain sleepy. They evacuated the building and we got to go home for the day.
IF YOU DO THIS - Make sure the container is out in an open area so YOU can get to it without being in an enclosed space with the Carbon Monoxide. Also, after she passes out, you need to keep the car running and the Carbon Monoxide levels up for a good period of time, at least 15-20 minutes, to make sure she is DEAD. Otherwise when you open up the chamber, she may revive on you!!!
My vet buddy is traveling and I can't get in touch with him right now to confirm the times, but I would say the longer the better. You don't burn much gas at an idle, so I'd let the car sit and run for 20 minutes or more.
Make sure that whatever you tie into the exhaust is not going to melt or catch on fire - a piece of laundry dryer foil/metal tubing would be good, or just connecting directly onto the car exhaust with some heavy duty aluminum foil held on with a piece of wire might work. You need a tube big enough for the air flow (basically similar in size to the exhaust pipe) and your Rubbermaid container needs to be fairly well sealed so that the Carbon Monoxide doesn't all leak out quickly. It's about the same density as air so it won't really "rise" or "sink" significantly in the box, but it will "leak".
After the 20 minute+ time frame, turn off the car and just leave the whole setup alone for a while so everything will cool off (tailpipe) and give some time for the box to leak out some of the Carbon Monoxide, I'd say at least 1/2 hour. This puts your total chicken exposure at close to an hour. Again, this is all to be done in an OUTSIDE/OPEN area, not in your garage!!!!!! When you are ready to open the box, open it and again walk away for a while - you really don't want to get a big whiff of this stuff yourself.
If you start feeling a headache or dizziness at any time, you are probably being exposed yourself. Your chicken might already be passed out or dead, but MOST IMPORTANT is that YOU are not doing the same!!!
My husband and I did learn to process roosters, so we HAVE cut the head off a living bird, so if we ever have to do this (put one out of her misery) we can do it without the "gas chamber".
If you do decide to try this option, plan it out carefully and BE SAFE.
DISCLAIMER - This method DOES produce a toxic gas that can kill a chicken OR another pet or a human being. It must be done with some basic safety care to avoid injury or death. I am not a professional chicken-gas-chamber-administrator, nor do I play one on TV, and I have also not stayed at a Holiday Inn Express for a while... I've never actually used this method, either, but I have been assured by a veterinarian that it is "peaceful". Based on my personal experience at my old lab, I would have to agree - the guy walking down the hall passed out pretty peacefully...