don't go above 3ppm with ammonia, you will grow different bacteria than what you want. I think a cycle being stuck with not enough ammonia is a misnomer, I recently cycled the ammonia out of a tank on accident simply by using dechlorinator without knowing that my water had chloramine in it (ammonia source). It never went higher than .25 ppm (that I know of, I wasn't testing so it could have been higher, but there was no meaningful ammonia source so I doubt it).
more often, a cycle gets stuck due to too-high nitrite levels. nitrites above 2ppm (I am kind of making this number up) can potentially cause a cycle to take longer than it would otherwise. If you are seeing high nitrites (I know the purples all look the same, dilute your sample by 50% with tap water, and then another 50% [before adding the test solution] etc until you get a reading you can decide what the color is), and then do water change(s) until you bring down the nitrites to something like 1ppm and see if your cycle picks up.
I very much applaud you for your patience by the way. Three months is no joke. Do you have access to a bacterial starter like Tetra safe start plus or seachem stability? I know folks in canada may need to order some of these online, and the tetra safe start plus might not like the cold temperatures (from what I understand, stability is spores instead of live bacteria so it is more resistant to temperature fluctuations, but that is also why it works slower). That could help speed your cycle along.
Otherwise, you are doing everything correctly, keep adding fish food to keep a bioload ongoing (if you are past the first stage of the cycle and stuck at the nitrites, then you might not be able to measure the ammonia after a while, just keep adding small amounts of fish food 1-3x/week to keep the bacteria alive). If your nitrites get too high (personally, I'd say above 1ppm, but I don't think there's any good data on what level of nitrites stalls the cycle), then do water change(s) to get them lower, and keep waiting. Try to get your hands on a bacterial starter so you can get to the fish part of the fish hobby as well