Nitrites extremely high - won't go down
I started cycling a new aquarium about five or six weeks ago. No fish in there yet. I was using household ammonia and everything was going great. A couple weeks in, the ammonia started being processed very quickly. I'd put in 2-3ppm and it would be down to 0 by the next day. I was "feeding" the ammonia-eating bacteria every few days while waiting for my nitrite levels to go down. My nitrite and nitrate levels have been maxed out for a couple of weeks now and haven't gone down at all. I started to suspect that the nitrite levels have gotten too high - when I test it's showing that they're at 5ppm but I think they're probably much higher.
I read a bit and saw some folks who said that it might be a good idea to do some water changes. At this point, I've done three or four 50% water changes over the course of a week and the levels haven't dropped. I also haven't been adding more ammonia because I don't want any more nitrites! But now I'm getting worried that I'm going to "starve" the bacteria that are present.
Any ideas what might be going on and what I should do? I have had high nitrate levels for weeks now, which makes me think that at least some of that nitrite is being converted. I just don't understand how it got so high and why it's not going down.
I thought about buying some Seachem Prime but I don't want to mess up the cycle.
Also, I tested my tap water for nitrites and it's showing up as 0ppm.