nitrite sky high but tetras spawnning

Hello, I am not necessarily new to aquaria or to this forum (just forgot my old login stuff). I have been doing the fishless cycling thing for a month now and finally added some fish to my 29gal. Three weeks ago my ammonia spiked off the charts, I did water changes to bring those levels down, it worked. Ammonia finally dropped to readable amounts and a few days later cleared up. I added a couple of shrimp pellets and some java moss from an established tank and my nitrItes spiked REALLY high (I cant say how high because it is over the readable scale) I have been doing 50 percent water changes for over a week and there has been no change.
Yesterday I added three black tetras and four julii cories. NitrItes are still high and I am doing 50 percent changes at least once a day but here is the weird thing, my black tetras are trying to spawn the female is huge with eggs and she is initiating mating play with the males.
I realize I should probably not added fish under these conditions but they seem to be doing fine.
I am using Aquarium pharmaceuticals tap water conditioner in the tank which seems to be nothing more than sodium thiosulfate.
I hope this answers all of your questions. yes the test kit works also.

sorry this is so convoluted

tank info:
filtration penguin 170 with carbon in the filter
population
4 julii corydoras
3 black tetras

ph 7.0
ammonia 0
temp 76f
Nitrites >5

thanks I figured that there was not much else I could do but I just wanted to make sure.

I am just doing water changes at this point I am going to add a little gravel from a friends tank to help kick start this whole thing. I really do appreciate your input though and will keep everyone posted as to my progress. My goal is actually to get the Corries to breed the tetras were just filler fish so I won't work too hard at protecting their eggs. Maybe in a year or so I will actively try to spawn the tetras. Thanks again.