I guys, just thought I should weigh in on this one. First off I want to extend my deepest sympathy to you for having to go through with this. This is something that I wouldn't wish on anyone. I am a biologist with extensive experience in humanely euthanizing animals. That being said, I have little experience working with birds, and have only been a chicken-owner for the last two months. This is certainly more difficult for you than it will be for them. Your girls are sick and hurting, as long as their death is quick and painless they will be better off. Regardless of this, I would strongly suggest finding someone else to do this for you. It is one thing to rationally know that something is right, and another to willingly kill off animals that you are emotionally attatched to.
In terms of the correct method, cervical dislocation, when performed correctly is entirely humane, and a vetrinary-accepted method of euthanizing an animal. We are only asked to follow up the cervial-dislocation with a "secondary-method" to ensure that the animal is dead. I have had some difficulty with cervial dislocastions in the past, and have ended up paralyzing animals without killing them, and as a result I rarely attempt this on my own. Chopping off the animals' head is also a commonly accepted ethanization procedure. It is gross, but it is extremely fast and efficient. What I worry about in this situation though, is that animals have an accute sense of smell. While the first couple of chickens may be naieve to what is about to ensue, it is unquestionable that the others will smell the blood and become nervous. Personally I aphixiate animals with CO2 gas, and while it is not necessarily the best method, it is what I feel most comfertable with. Even in a lab setting it takes ~1 minute for the animals to go down. Death is painless certainly, but the animals are gasping for air until they fall asleep.
As for the mycoplasm, most everything I have read here is correct to my knowledge. Mycoplasm is a tiny bacteria that is not visable under a standard light microscope. And yes, it is generally present in the enviornment, but we certainly try to eliminate it our laboratory setting, as it slowley and progessively interferes with the vitality of cells. Here in the lab, any cells that have become infected with mycoplasm are discarded, regardless of the cost/time involved with the sample. It is simpley not practical to try and fight off mycoplasm at the risk of contaminating other cell lines.
I don't know how this helps you, but please feel free to contact me with whatever questions you may have.
hang in there