I've thankfully never had to euthanize a fish (knock on wood) but I have read up on the subject, and there are some commonly used methods out there that are revealed to be shockingly inhumane... :(
One common one is the freezer method - placing the fish in a ziploc bag of tank water and putting them in the freezer. While a good idea at first glance, you have to consider the reality - because of the cold-blooded nature of fish, a decrease in temperature like this will SLOW DOWN instead of completely stopping bodily functions, thus leaving the fish still alive as cell crystalization and freezing begins. What a horrible and agonizing death that would be. The same goes for things like flash boiling. I had no idea that these methods were this heinous either until I read this article:
It details acceptable and unacceptable methods of euthanasia for fish. It seems that the number one acceptable method is immersing the fish in a massive overdose of veterinary grade anesthetic.
Personally, I agree with AngelicScars, and I don't think I could ever bring myself to euthanize one of my fish - but it some cases, it would be inhumane of me not to.
I hope this is helpful to you in making your decision, and I'm SO sorry that you have to be making a decision like this at all. :( Your fish is very lucky to have such a caring owner! :) Best wishes (and as I do with all animals, I'll be keeping my fingers crossed for a miraculous recovery!)