severing the spine is IMO the quickest way if you have a steady hand and do it quickly. This is going to be a morbid paragraph and it is not my intent at all to make anyone uneasy. I have used a very sharp pair of scissors on a small fish that had become ill and was not squirming. Your
eel is much larger and a meat cleaver may be the quickest way to do it. Not pretty but quick and as I understand fish neurology relatively painless. THe technique of freezing is considered humane by most. A third technique is to place the specimen in a small container, add alka seltzer tablets and cover. Co2 will saturate the container causing a narcotic effect, and then resperitory failure and death. This technique is also considered humane. I am very sorry you have to make this decision but I thought I could at least give you some options.