I kept a Gymnothorax Undulatus (undulated Moray) for a few years (25 years FWIW) so I may be of some assistance.
As others have asked, to help determine the problem and it's cause, we need more information. Tank size, how long it's been set up, lighting (eels don't really like too much light, making them less suitable for a reef aquarium), filtration (live rock/bioballs/etc), other tank mates, and current parameters. The ones that count for the Eel are Ammonia, Nitrite, Nitrate, pH, salinity, and temperature. The Eel doesn't care about Ca, Alk, or Mg. Has the tank been treated with copper or any other medication? Smooth skinned fishes like Eels can be especially sensitive to copper. If it has, running a good amount of carbon, preferably in a reactor, will help to remove it and any other toxins in the water.
Pictures will help immensely.
Once we can figure out what is going on we can think about treating it properly. Medicating for the sake of medicating can do more harm than good so until there is a diagnosis I would suggest you not try to treat what might or might not be the problem.
As for actually giving it the medication, as long as it is still eating it is really pretty simple. Add the meds to it's food rather than trying to treat the entire tank. It will take much less of whatever meds needed and will have a much greater chance of success.
Eels are very resilient creatures. I gave mine up for doomed several times over the years and each time it recovered just fine. It's longest hunger strike was nearly 6 months. The first time it escaped, I found it dried up on the floor but still hanging on. I dropped it into the tank, and other than a really bad bacterial infection was none the worse for wear.
