It is best to remove all livestock to prevent the likelihood of your other tank mates from getting the ich. Eels are not likely to get ich which is a good thing, though if you have livestock death, then the elevated ammonia and nitrites could prove fatal to your eel.
Having said this, unfortunately, you need to let the ich cycle take its course which usually lasts anywhere from 7 days to many weeks based on water temperature. The ectoparasite cycle tends to speed up under elevated water temperatures plus minimizes the chances of reproducing though, this can bring about reduced dissolved oxygen in your water column. Their are certain times during the life cycle of this parasite when you can treat it effectively through the use of Formalin and hyposalinity(dropping the salinity level) or copper treatment plus garlic also helps, just to name a couple. A word of caution...IF You remove your livestock from your main tank and then reintroduce them into your main tank again without making sure the ich cycle has completed or the parasites have died he to no host, your fish WILL get the ich again.