The substrate should have quit producing ammonia, or be producing such a low level that the bacteria should be able to handle it. Since the bacteria are not handling it there are 2 possibilities:
There are other sources of ammonia, and the bacteria have not grown to a big enough population to deal with it.
Something is preventing the bacteria from growing.
API Stress Zyme does not contain the right species of bacteria to boost the nitrogen cycle. Look for Nitrospira species. Most other bacteria in a bottle are decomposers, or are bacteria that die out in just a short while.
Yes, dead plants add ammonia.
Poor filter performance is hindering the growth of the nitrogen cycle bacteria. They need high levels of oxygen. The substrate may be contributing to the problem, too. If I remember, that is one that removes the carbonates from the water. These bacteria need carbonates.
What I would do:
I would get the appropriate Aquaclear if you want a HOB filter. Get one that moves the volume of the tank 10 times per hour. A 10 gallon tank would get 100 gallons per hour. if you want a canister I have had decent service from the Rena Filstar product line. You could get by with less gallons per hour, perhaps as little as 5x, but I usually size these at something close to 10x, too.
Do another water change to get the ammonia as close to 0 ppm as you can. Clean out all the plants that are not making it, and trim those that are looking poor but can recover. High ammonia can burn the leaves.
Add the right dose of any Nitrospira species of bacteria.
Allow the ammonia to rise, and monitor ammonia, nitrite and nitrate.
Also, monitor the KH. If it is under 3 German degrees of hardness add enough baking soda to keep it at a minimum of 3dKH.
If the ammonia and nitrite fluctuate for a few days then hit and hold at zero, then the Nitrospira bacteria are doing their job, but they will starve without ammonia. Until you get your livestock feed them ammonia. Add enough ammonia to bring the test to 3 ppm once a day. If this is too much for the plants then add only enough to read 1 ppm, but do this twice a day.
When the fish arrive do a big enough water change to bring the nitrate down really low, then add the fish.
If the Nitrospira you buy is dead, then continue caring for the tank while doing the fishless cycle.
Here is the fishless cycle.
You too can boast that "No fish were harmed in the cycling of your new tank"
Cycling a tank means to grow the beneficial bacteria that will help to decompose the fish waste (especially ammonia). These bacteria need ammonia to grow. There are 3 sources of ammonia that work to do this. One is fish. Unfortunately, the process exposes the fish to ammonia, which burns their gills, and nitrite, which makes their blood unable to carry oxygen. This often kills the fish.
Another source is decomposing protein. You could cycle your tank by adding fish food or a dead fish or shellfish. You do not know how much beneficial bacteria you are growing, though.
The best source of ammonia is... Ammonia. In a bottle.
Using fish is a delicate balance of water changes to keep the toxins low (try not to hurt the fish) but keep feeding the bacteria. It can take 4 to 8 weeks to cycle a tank this way, and can cost the lives of several fish. When you are done you have grown a small bacteria population that still needs to be nurtured to increase its population. You cannot, at the end of a fish-in cycle, fully stock your tank.
The fishless/ammonia cycle takes as little as 3 weeks, and can be even faster, grows a BIG bacteria population, and does not harm fish in any way.
Both methods give you plenty of practice using your test kit.
How to cycle a tank the fishless way:
1) Make sure all equipment is working, fill with water that has all the stuff you will need for the fish you intend to keep. Dechlorinator, minerals for GH or KH adjustments, the proper salt mix, if you are creating a brackish or marine tank. These bacteria require a few minerals, so make sure the GH and KH is at least 3 German degrees of hardness. Aquarium plant fertilizer containing phosphate should be added if the water has no phosphate. They grow best when the pH is in the 7s. Good water movement, fairly warm (mid to upper 70sF), no antibiotics or other toxins.
2) (Optional)Add some source of the bacteria. Used filter media from a cycled tank is best, gravel or some decorations or a few plants... even some water, though this is the poorest source of the beneficial bacteria.
Bacteria in a bottle can be a source of these bacteria, but make sure you are getting Nitrospira spp of bacteria. All other ‘bacteria in a bottle’ products have the wrong bacteria. This step is optional. The proper bacteria will find the tank even if you make no effort to add them. Live plants may bring in these bacteria on their leaves and stems.
3) Add ammonia until the test reads 5 ppm. This is the non-sudsing, no surfactants, no-fragrance-added ammonia that is often found in a hardware store, discount stores, and sometimes in a grocery store. The concentration of ammonia may not be the same in all bottles. Try adding 5 drops per 10 gallons, then allowing the filter to circulate for about an hour, then test. If the reading isn't up to 5 ppm, add a few more drops and test again. (Example, if your test reads only 2 ppm, then add another 5 drops) Some ammonia is such a weak dilution you may need to add several ounces to get a reading.
4) Test for ammonia daily, and add enough to keep the reading at 5 ppm. You probably will not have to add much, if any, in the first few days, unless you added a good amount of bacteria to jump start the cycle.
5) Several days after you start, begin testing for nitrites. When the nitrites show up, reduce the amount of ammonia you add so the test shows 3ppm. (Add only half as much ammonia as you were adding in part 4) Add this reduced amount daily from now until the tank is cycled.
If the nitrites get too high (over 5 ppm), do a water change. The bacteria growth is slowed because of the high nitrites. Reducing the level of ammonia to 3 ppm should prevent the nitrite from getting over 5 ppm.
6) Continue testing, and adding ammonia daily. The nitrates will likely show up about 2 weeks after you started. Keep monitoring, and watch for 0 ppm ammonia, 0 ppm nitrite and rising nitrates.
7) Once the 0 ppm ammonia and nitrites shows up it may bounce around a little bit for a day or two. Be patient. Keep adding the ammonia; keep testing ammonia, nitrite and nitrate.
When it seems done you can challenge the system by adding more than a regular dose of ammonia, and the bacteria should be able to remove the ammonia and nitrite by the next day.
If you will not be adding fish right away continue to add the ammonia to keep the bacteria fed.
8) When you are ready to add the fish, do at least one water change, and it may take a couple of them, to reduce the nitrate to safe levels (as low as possible, certainly below 10 ppm) I have seen nitrate approaching 200 ppm by the end of this fishless cycle in a non-planted tank.
9) You can plant a tank that is being cycled this way at any point during the process. If you plant early, the plants will be well rooted, and better able to handle the disruption of the water change.
Yes, the plants will use some of the ammonia and the nitrates. They are part of the nitrogen handling system, part of the biofilter, they are working for you. Some plants do not like high ammonia, though. If a certain plant dies, remove it, and only replace it after the cycle is done.
10) The fishless cycle can also be used when you are still working out the details of lighting, plants and other things. If you change the filter, make sure you keep the old media for several weeks or a month. Most of the bacteria have been growing in this media (sponges, floss etc).