Do not add ammonia for now and drop the temp to about 80F. But lets try something here.
Go to a local supermarket or all purpose pharmacy type store and grab a gallon of distilled water. Not mineral or spring, distilled.
You need a clean measuring cup and a clean glass. Collect from the tank 1/2 cup of tank water. Add to this 1/2 cup of the distilled water. Now use this mixture as the water for testing for nitrite. If you get a reading under 5, multiply it by 2 to know the actual ppm. Then in 2 days add a 1 ppm snack and start testing nitrite daily. Ammonia should not read for long so don't bother testing it for now. Just follow the ending directions for cycling and doing the 3 ppm cycling tests again. However, if the reading is still off the charts (a solid 5.0), do the following.
Pour out the water in the measuring cup into the glass and then return enough to be 1/2 cup. To this add 1/4 cup of distilled water. This means you will have 1/3 tank water and 2/3 distilled water in the cup which will be 3/4 full. Now do a nitrite test using this water. If the result is under 5, multiply it by 3 to know the actual ppm in your tank. In 3 days add a 1 ppm snack and begin testing just nitrite daily. Follow the ending directions for cycling and doing the 3 ppm cycling tests again. However, if this test is still off the charts (a solid 5.0), do this.
Change as much water as you can. After refilling, test for nitrite. It should be under 5. If not change more water to get it under 5. Then wait for it to drop. Test every other day for nitrite. If the 2nd time you test ammonia is 0, add enough for 1 ppm as a snack and then keep waiting and now test daily. Ammonia should vanish fast and when nitrite hits .25 ppm or less, do the 3 ppm ammonia addition test. You will be at a point in the directions where you can follow them again as written.
The reason for all this is I would like to be sure your nitrite is not over 16 ppm. If it is it is messing things up. If it is not is should settle out in some number of days.
Incidentally, if you check the directions they say when you added to 3 ppm and got that 0/0 and did the water change, the next step was to add fish. Instead you added 3 ppm again. Next, using media from a cycled tank changes how you are supposed to do the cycle. Seeding a tank reduces the time involved. In the Suggestion part of the article it says:
If you can add gravel or filter media from a healthy cycled tank, it will accelerate the process. The more you can add, the more it will help. If you can do this, reduce the time between testing from every 3 days to every 2 days. Test on days 3, 5, 7, 9 etc. Then reduce the every 2 day testing to every day.
Did you do this? Also did you keep a record of your test results and ammonia dosing amounts (include how many ml you added) and when. Also, what strength ammonia are you using? It is hard to try and diagnose where a tank is during a cycle without knowing how it got there.
If you think about what you have reported it is basically this:
Started to cycle with some additional bacteria. Dosed ammonia etc.
Saw ammonia drop to 0 indicating enough ammonia bacs present to handle ammonia.
Saw nitrite rise and then drop to 0 or almost 0, indicating enough nitrite bacs present to handle almost all the nitrite.
Dosed ammonia to test if the tank was cycled. Got 0/0 readings in 24 hours indicating the tank was cycled.
All of this was perfect. Your tank showed as being fully cycled You changed the water, not something that would kill only nitrite bacs and not the ammonia ones or remove either. Cleaning the glass would not have this effect either. Most of the bacs are in your filter and on the gravel and decor. They, and other tank bacs, all live in the same bio-film they create. What wipes out one also harms the other. So if the ammonia bacs are still there, what happened to all the nitrite bacs that were also there? Your tank was cycled, you changed the water and the tank was cycled for ammonia but no longer cycled for nitrite. Not possible. Something is fishy.
But what is important is getting things back on track and finished and the above nitrite testing should do that along with the other info I asked for. Of course before going out and getting the distilled water etc., test for nitrite, maybe it has dropped back under 5. If it has, dose the snack in 2 days and when 0/.25 cross fingers and do the 3 ppm again.