Not to be mean here but what people said about 13 gallons being way too small for your pet is absolutely 100% correct. It probably has no nitrogen cycle and is almost definitely the cause of the fin rot. For a big fish like that in such a tiny tank, you will be needing water changes of 50% like every day as the toxic ammonia and nitrate levels rise. I bet the shop you got the fish from gave you very bad and outdated information on how to care for it or they just wanted to sell it since flowerhorns are expensive. You might want to lurk around on the site and learn about the basics of fishkeeping if you weren't told anything.
The good news is that most likely the fin rot can be treated and reversed easily, hopefully the fish hasn't had too much irreversible damage to the gills, if you really only changed the water every two weeks (who told you that haha? that's nuts) your fish might have ammonia burns too (also, does the fish have any wounds or is acting weird/breathing very hard?). A picture of your tank and the fish would be helpful.
IMO you should try aquarium salt first before the medicine since the fish is probably really weak and the medicine can kill weak fish if you dose too much too fast. (You can just use kosher salt without iodine in it since it's cheaper.) The proper salt dose is about 1 tbsp of salt per gallon. Keep up on the water changes while you do this. Replace the same amount of salt that you take out (on your tank if you do 50% it would be like 6.5 tbps replaced.) Just dissolve the salt in the tank water before you add it so the fish doesn't eat it. Cichlids can tolerate salt well so it should work. Change 50% of the water EVERY DAY until the fish shows some improvement. Make sure the temperature is the same as the tank water and you use your dechlorinator too. Reduce your feeding to a small amount once a day. Basically you want to treat your tank as a quarantine. I just had a convict cichlid rescue with the same issue and this worked pretty well to get him strong enough to go in my main tank.
For the tetracycline, I would use that too if the salt or water changes isn't working (the salt can take a few weeks to heal the fish, be patient). I haven't tried it myself though so i'm not sure on the water change schedule. Follow the dosage instructions on the medicine box and change the water everyday when in doubt.
PS. while you are treating the fish, please see if you can find something like a used 55-75 gallon tank and a hang on back or canister filter, lots of people give them away on craigslist and stuff for cheap or even free. It should hold you off until you can get something like 100 gallons, which is what that fish really needs. That would be way better for the flowerhorn if it is the only fish you have. If you can't treat it yourself or get a big tank you should rehome the fish to someone who can care for it right. If the fish is extremely sick you might have to euthanize it (maybe a vet can help do that if you don't want to), you don't want your pet to suffer needlessly.
