The quickest way to kill a small fish is to wrap it in a paper towel and hit it on the head with something hard. The drawback is it can be messy.
Clove oil mixed with a small amount of alcohol can be added to a small container of water containing the sick fish. You have the water aerated gently and add a drop of clove oil, wait about 20-30 seconds and see how the fish responds. If it continues to swim you add another drop and wait a bit. Continue doing this until the fish stops swimming and sinks to the bottom. Then add a few more drops and pop the container of water in the freezer. Leave until frozen solid then bury the iceblock in the garden.
If you can't get clove oil (available from most chemists/ pharmacists or online) and you can't bring yourself to hitting a fish on the head, put them in a small container of aquarium water and put a lid on it. Put it in the freezer and walk away. Bury the iceblock after it's frozen solid.
Freezing isn't the nicest way to kill them but it's the easiest for most people.
Clove oil is good if you can get it.
The following links are about euthanising/ sedating fish and might interest you.
Ammonia, nitrite and nitrate are all forms of nitrogen and are all toxic to fish and other aquatic organisms. Ammonia is the most toxic (especially in water wiht a pH above 7.0), followed by nitrite (more toxic in water with a pH below 7.0), and then nitrate.
Ammonia and nitrite have to be kept at 0ppm at all times.
Nitrate should be as close to 0ppm as possible and under 20ppm at all times. Nitrates do cause long term damage to fish health. Whereas ammonia and nitrite tend to kill fish quickly.
BASIC FIRST AID FOR FISH
Test the water for ammonia, nitrite, nitrate and pH.
Wipe the inside of the glass down with a clean fish sponge. This removes the biofilm on the glass and the biofilm will contain lots of harmful bacteria, fungus, protozoans and various other microscopic life forms.
Do a 75% water change and gravel clean the substrate every day for a week or until the problem is identified. The water changes and gravel cleaning will reduce the number of disease organisms in the water and provide a cleaner environment for the fish to recover in. It also removes a lot of the gunk and this means any medication can work on treating the fish instead of being wasted killing the pathogens in the gunk.
Make sure any new water is free of chlorine/ chloramine before it is added to the tank.
Clean the filter if it hasn't been done in the last 2 weeks. However, if the filter is less than 6 weeks old, do not clean it. Wash the filter materials/ media in a bucket of tank water and re-use the media. Tip the bucket of dirty water on the garden/ lawn. Cleaning the filter means less gunk and cleaner water with fewer pathogens so any medication (if needed) will work more effectively on the fish.
Increase surface turbulence/ aeration to maximise the dissolved oxygen in the water.
Start a new thread in the emergency section of the forum and post clear pictures (and maybe video too) of the fish so we can check them for diseases.