If it's been there for two weeks, it's probably breaking down, and could be really bad for the fish.
I agree that stress coat is a good thing.
Depending on the size of the eel, you may want to get yourself a pair of heavy-duty gloves (I would go with a pair of knit gloves, topped with a pair of the cut-resistant kitchen gloves, topped with a pair of
PVC-dipped gloves), grip it immediately above or immediately below the gills, and use a pair of forceps to debride the gills as best you can (be very careful not to damage the gills themselves).
If PimaFix/MelaFix are aquarium safe, you may want to use one of them to limit the growth of fungus/bacteria.