I can kind of follow your description, but when you "stocked the tank", how many additional fish did you add beyond the two for your initial cycle?
Ammonia --> Nitrite --> Nitrate ... only thing that significantly reduces nitrates for your freshwater system is water changes.
How does the conversion from one to the next happen? The thanks belong to bacteria!
Certain bacteria consume one and produce the next as a bi-product. When you have a steady amount of fish producing a certain amount of waste, the bacteria build up to a level where they can sustain on the given quantity. if you add fish, suddenly there is more waste, so both ammonia and the nitrites will have individual spikes as the bacteria colony needs to grow in order to consume the newly increased amount of waste being produced. Thus you have a "mini-cycle" of sorts that occurs.
FWIW, there are some filter pads that absorb or neutralize ammonia and/or nitrites. These are a crutch of sorts because doing so will not allow the bacteria colony to grow to a suitable level for a tank to become self-sustaining with regards to water quality (self-sustaining with exception to necessary water changes).
Another, better option to cut down the nitrite levels if you find them too high is through the use of water changes. 50% water change --> 50% reduction in nitrite level (assuming there's not some inherent level of nitrites in the supply water). Doing this will lengthen the cycle, but having a longer cycle a better alternative to having fish dying due to heightened levels of ammonia/nitrites.
