I have no idea. There's something wrong. My fishless cycle completed a few days ago. It took 32 days, but by two weeks it was processing ammonia overnight to zero. Nitrites stayed way off the charts most of the last 3 weeks, but it finished. Last time I cycled a 10g it took even less time (but I did seed it).
People do this all the time without issues, even if impatience makes it seem like there are issues. You are almost too patient; there's something more wrong here.
Are there any really knowledgeable LFS's nearby that you can get to test your water, compare to your tests? I can't see how the test can be wrong, but maybe there's a bad kit, and you are getting erroneous readings. Did you test the tap water (maybe you said, don't have time to read the whole thread right now); bad readings there might indicate either bad tap water or bad test kit. A bad container of prime (does it have a strong sulfur smell (it should).
Frankly if it were me at this point I'd be tempted to buy an aquarium screen saver.
More seriously I think I'd start over at this point. All media in the trash, clean the tank thoroughly without soaps, but scrape off any residue that might be leaching... look for something that might be contaminating it.
But the other aspect is see if a local LFS or club might have someone who can give you some seed material (pieces of sponge, etc.) that are from a cycled tank, that should speed things up. But I just did my 30 with zero seed, zero bottled bacteria... nothing but pure water in a clean tank. Bacteria is everywhere, it just takes longer to start without a seed.
Sorry... maybe others have a possible explanation. I'm out of ideas.