TL;DR - This is completely normal. You're exactly where you need to be.
Why I say that:
Over the past 8 months I've started and cycled 10 "aquariums" and your experience closely matches mine with the first. That one I stared with Dr. Tim's bacteria and Fritz Ammonium Chloride. It did much like yours and wiped the ammonia out in a week and nitrites about a week later. This was my first tank in years and I did not dose again. We added fish and that night the world caught on fire and a power outage stopped my canister filter and nearly killed the entire cycle. There was enough, and I didn't lose any fish, but I did fight high nitrites for a week or so. You're doing well by dosing multiple times.
About 12 years ago, I took the advice of the LFS owner and started an Mbuna tank with Eco Complete cichlid sand (Sahara sand- very pretty, still like it). Like you, all brand new gear, but, plus fish immediately. 6-10? I can't remember. While I would never do this now after having learned so much, the thing is, there were no ammonia problems, so I'm guessing that the live bacteria in that bag actually do something.
Now, the most funnest part - cycling the K1.
In prep for the big Malawi Reef, I did not want to wait for the tank to cycle, so while I was waiting for the tank to be built, I started the cycle. I got two feet of K1 (Smoky Mountain Bio Media) and got one foot of it cycling with powerheads in a rubbermaid container in the garage. I put it under water, added ammonium chloride (see above. I bought the Fritz container, so I now have an infinite supply of the stuff), squeezed a few cycled sponges in there from another tank (by this time I had 6-7 and no lost fish or ammonia/nitrite issues) and let it run fluidized. I also had some cycled sponges and pot scrubbies in this box. It took a few days for the oxidization to complete. K1 is very slow cycling, especially fluidized, I'm told, and my experience bears this out.
I kept dosing and after about 2 weeks I needed the pot scrubbies and sponges for yet another tank I was starting, so I took them out. At this point the K1 REALLY slowed down. After the first dose, it took a week to clear. I was dosing to about 4 ppm. I realized that the media was seeded, but the bacteria on the pot scrubbies had been doing all the work. I continued dosing to grow the bacteria, and got it down to where it could go through 4ppm overnight, so I started upping the ammonia. This is when the nitrites started hanging worse. It was much like what you described. Ammonia was gone overnight, but it took 2 more days for the nitrites to go away.
Since I had no line of sight on the tank, I decided to test things out and just really mess the media up and get it growing and cycled. I started adding lots of food for the media to eat. I'd dump a bunch of cheap fish food in there every few days. I added a few tablespoons of SUGAR to aid in the creation of bioflocs and about a teaspoon of BAKERS YEAST. Yup, you read that right. OVER NIGHT the nitrites dropped to zero.
I upped the ammonia I was dosing steadily. I bought a scale, to be more accurate, and I was dosing about a half gram per day. At this level, the API test was dark green daily. I started going to 8 ppm, then I stared just messing it up with ammonia. The test tube would be a color I referred to as meaninglessly green. Within 24 hours, it would be clear. So, after about two months of this, I decided to really see what it could do. I dosed 4.5 grams of Fritz ammonium chloride. This is enough to bring 100 gallons of water to about 4ppm. In 20 gallons of water in the rubbermaid tote, it turned the API tube black. You could not see through it. In the garage, you could smell the ammonia. It took 3 days to eat this amount of ammonia. Based on how fast nitrates built in my quarantine tanks, I would say it would take about 500 juvenile Mbuna about a week to make 4.5 grams of ammonia.
This means basically, that overdosing ammonia won't do anything to the cycle. Mess it up. Enjoy! Nature is flexible. What it WILL do is piss you off. The reason for this is that the tests are not meant to measure changes at these levels. That's why anything over 6ppm to me is "meaninglessly green" you find the same thing with the purple color on the nitrite test. If you don't dose to these levels, you'll more easily be able to see what's going on with the tests, so it LOOKS like nothing is happening, but it is. False negative.
The other thing that I should point out is that my tap water has very little buffering capacity. Therefore, after a few doses of ammonia, the box would build nitric acid, lower the PH and shut the cycle down - this could also be another reason that the folks say to not dose so high. It actually can slow the cycle down if the PH drop is not anticipated and corrected for.
I may have missed it, but what's the volume of water you're playing with?