If you want to clean the filter media, then do so in dirty tank water (ie some of the tank water from the tank), never use tap water as this will kill any beneficial bacteria and just lightly swish it around in the water, you dont want to lose the bacteria formed in the sponges, just get the gunk off to improve the flow.
You are seeing nitrItes rising because you are in a cycle, the amount of waste the fish are producing is ammonia, this will continue to rise until there are enough Abacs to convert the ammonia to nitrite, you obviously have abacs or you wouldnt be seeing an nitrIte, however for every 1ppm of ammonia, I believe the conversion is to 2.47ppm of nitrIte, so the more ammonia you have, the higher the nitrIte will be, until eventually the Nbacs grow enough to convert the nitrIte to nitrAte (the end of the cycle process). Each 1ppm of nitrIte you have is converted to 4.? of nitrAte (think thats the approx conversion). Either way, you are cycling and the ammonia will rise, then the NitrIte will rise until all the bacteria are established enough to do this sufficicently so you dont actually see any ammonia or nitrIte as they are converted to the end of the process nitrAte.
You will need to do several large water changes daily to get the ammonia and nitrIte down to below 0.25ppm whilst in the cycle for the safety and health of the fish. If this means 5x water changes each day, then Im afraid thats what you have to do, you will be doing this for several weeks until you see no more ammonia or nitrIte, which is when you will have finished the cycle.