No nitrites but nitrates still high

That means your bacteria are efficiently breaking down the ammonia to nitrites to nitrates. You should do a water change, add a bit more food, check for a spike in ammonia or nitrites in the next couple days, If none, then your tank is cycled, do another water change, then add 1 fish.
 
Unlike ammonia and nitrite, bacteria doesn't break down nitrates which are the end product of the nitrogen cycle. You need to do water changes to get your nitrate levels down. Sounds like your cycle has finished and as BJ says do a water change and then you can add your 1 fish and your clean up crew!
 
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