It's probably my explaining, I feel like it would be easier with props
Imagine you have a cup of milk, pour out half the milk and top up with water, you'd agree I'll now have 50/50 water and milk - right? If you pour out half of that 50/50 milk/water cup top it up with water again - do you now have a glass of pure water? After all that's two 50% changes ... so you'd think that means now you have 100% water? Nope, you still have milky water!! (Feel free to try it
) To be accurate you now have 25% milk and 75% water, because when you poured out half the glass the second time you were pouring out some milk and some of the new water, not just milk. If you kept going you'd eventually dilute it until you couldn't detect milk. On the other hand, if you poured out all the milk in one go and then refilled with water then you would have 100% water. That's why two 50% changes don't = a 100% change.... and a 90% in one go change makes a 90% change. You'd need to do 3-4 50% changes to achieve the same dilution as a single 90%.
1st 50% change = 50% new water
2nd 50% change = 75% new water
3rd 50% change = 87.5% new water
4th 50% change = 93.75% new water
Now, to make it worse, imagine the water you are refilling with already has some milk in! Just like your tap water already has nitrates.
If your tank has 10ppm and your tap has 5ppm your numbers will look like this:
1st 50% change = 7.5ppm
2nd 50% change = 6.25ppm
3rd 50% change = 5.625ppm
That's assuming no more nitrate is generated, when we know it is as it's raising from 5 to 10ppm in 3 days. Assuming your water change is every 3 days that makes your number look like:
1st 50% change = 7.5ppm (raising to 12.5ppm)
2nd 50% change = 8.75ppm (raising to 13.75ppm)
3rd 50% change = 9.37ppm (raising to 14.375ppm)
4th 50% change = 9.6875 (raising to 14.6975ppm)
5th 50% change = 9.84375 .... and so on
As you can see it stabilises out just under 10ppm. It's physically impossible for you to water change your nitrates lower.
I apologise for all the maths, which is a bit off topic to your original question
10ppm nitrates shouldn't cause the cyano though, it's not a terrible number. My water comes out the tap at 40ppm. If you wanted to try and reduce it, and it wouldn't be by much (we're talking the difference between stabilising around 5.3ppm v. 9.8ppm) your best bet would be floating plants or plants with the roots in the water as they grow strongest with access to CO2 in the air. A regular planted tank would be nice, but I wouldn't count on it making a significant difference to nitrates. And, if you do drop them you'd end up needing to add them back in to keep the plants healthy.
The fact you saw the cyano die off with a five day blackout is great, I think that is your cure. My advice is to physically remove as much cyano as you can, use a tooth brush and a syphon, take anything out you can to scrub. You want the tank to look cyano free. Then do a five day blackout again, making sure to cover the tank to prevent ambient light. Fingers crossed once you've done that it won't reappear. Lowering the light following should help make sure. I've successfully cured it that way myself.