Question: Tank not cycling

Ok, so I have a problem. I have been cycling my tank for a little over a month, and nothuing seems to be happening. My ammonia is at 1ppm right now, but my nitrites keep going really high (5ppm), and it happens almost the instant I add more ammonia, and it doesn't go down on it's own. Sometimes the nitrite spikes without the ammonia levels even lowering. I have been doing water changes, but I can't seem to get the cycle started. I have been seeing nitrates from almost the first week of cycling as well, which I have since heard is unusual. I have been using water dechlorinator (Prime) with every water change, but I have no idea why this is happening? How long should I leave my nitrites at 5pppm before changing it? I usually let it sit for a day or two to see if they will convert to nitrate, is that okay?

I dont have experience with cycling aquarium with chemicals. But have you tested your treated tap water for ammonia, nitrite, nitrates? If you can get a piece of used filter from cycled tank. It could take as little as 2 days in 10 gallon. Some pet shops will sell them.

I've tested my water before, there is no ammonia or nitrite. Sometimes my cat tries to drink out of my tank, could that be causing a problem?