Thank you for waiting. Since no pine coning, we can rule out dropsy. Your koi is producing excessive slime coat. That can be caused by a number of different illnesses and parasites. These same conditions can cause abdominal swelling and breathing difficulties. What is odd is that only one koi is affected. Usually more than one will come down with symptoms at the same time. The very best thing to so, especially of this fish is important to you, would be to consult an aquatic vet. The vet can take a skin scraping to examine under a microscope. That is the best way to get a definite diagnosis. If you want to do that, this link will take you to a directory of such vets. You'll need to check "private practice' and 'ornamental aquatic animals' in order to get the proper list.
If a vet is not an option, you can either try to treat the fish in the pond or move him to a hospital tank. An extra large Rubbermaid tote or a plastic wading pool would work. If you use a hospital tank, fill it with water from the pond and place it where it will be protected from predators. Provide high aeration. Whatever is wrong with the fish seems quite advanced, and sometimes strong medications can make a situation like that worse. Whether leaving the koi in the pond or in a tank, the first step is to use aquarium/pond salt. It is the safest medication, and salt can kill some parasites, and it helps fish cope with any sort of stress. Use one teaspoon per gallon of water. There are approximately 125 teaspoons per pound of salt.
If you are able to move the fish to a hospital tank, you can also feed him a medicated food. These will treat most bacterial illnesses. The foods can be hard to find. Here is an online source:
Metronidazole is often used for parasites, but without a skin scraping we can't confirm parasites. If you have seen the fish flashing or rubbing on objects in the tank, those signs are good evidence of parasites. In that case, you may want to go with the metronidazole before the medicated food. I wouldn't use both at once. Metronidazole can be bought in pet stores and pond supply stores. If you see more fish developing symptoms, you'll need to treat the whole pond. A parasite called costia is among the most dangerous. It causes excessive slime production, and hanging near the water surface. It requires an even more aggressive approach.
A combination of malachite green and formalin is most effective. You may be able to buy this in a store that sells pond supplies, but it's more likely you'll need to order it online. The following merchant sells it in quantities suitable for big ponds:
I have given you what is probably a confusing amount of information. In summary, the first choice is to consult an aquatic vet. If you can't do that, treat with salt and medicated food. Only if you see some of the signs of parasites, skip the food and go with salt and an anti-parasitic medication. Unfortunately, with fish illness, we often have to experiment when we can't consult a vet.
If you have more concerns, just let me know. I hope your koi will be able to recover.