I am really sorry to hear about Lizzie not being well, and possibly dying. Many use the cervical dislocation or broomstick method for putting down a chicken. Some cut the jugular vein from which they will bleed out in a minute if hanging by the feet or upside down in a killing cone. There is the hatchet method as well for decapitation. Some use engine starting fluid that contains ether, an anesthetic, on a cloth in a large box. If your vet euthanizes her, they may also do a necropsy for you to look for what was wrong.