Some thoughts that I hope will help here:
Cycling a tank using Dr. Tim's One and Only and his bottled ammonium chloride is a great way to cycle a tank, but can take as long as 3 weeks or so in some cases.
One of the reasons it can take that long is because there seems to be a disconnect between the instructions on the bottle of ammonium chloride, and the instructions posted on his website, if I remember correctly. If you follow one set on instructions, you get fairly high ammonia, which takes longer to go to zero, as well as subsequent higher nitrites, and eventually high nitrates.
Do you know how high the ammonia got at its peak?
As others have said, you cannot test accurately for nitrates when you have nitrites present, so don't worry about your nitrites at this point (or ever, frankly, it's unimportant in these tanks). Nitrite is non toxic in marine aquaria, and many here at R2R don't test for it or worry about it at all. Here's some info on that if you get bored:
https://www.reef2reef.com/threads/nitrite-toxicity.402708/post-4805029
If you had ammonia spike to about 2.0 ppm or more, and then it is now (near) zero, and you're seeing "something" for nitrates, you can assume your cycle is proceeding appropriately.
You can test your system by adding enough ammonia chloride to get the ammonia concentration to ~2.0 ppm, then check again in 24 hours and if the ammonia is near zero, you're good

No reason to wait for nitrites to come down to zero, you're safe to start slowly adding some hardy livestock.
API test kits don't get a lot of love here on R2R, but honestly they are fine for cycling a tank. There's some things to be aware of, like the ammonia can show ~0.25 ppm when it's likely nearly zero. None of our hobby level kits will give you laboratory level results, so use the API kits to track trends; ammonia went up, and is now down. Nitrates were zero and are now rising. Don't expect hard numbers with these test kits
I hope this helps and good luck!