I've never had a turtle tank and I'm speaking from my fish-tank experience. Yes, everyone's got that problem. In fish-tanks algae is a result of too much light, too much nitrates, nitrites & phosphates. Aquarists fight algae by frequent water changes, algae eating fish, dense planting and control of the three parameters. And the fight is tough

We even calculate the quantity of light per liter. In turtle aquariums the problems are too much waste + light. Too much waste means too much nitrates, nitrites & phosphates. Any food or excrements left in the water start to rot and turn into nitrates. Here come a very essential moment in aquariums - filtration. In the filter there are colonies of useful bacteria which transform the nitrates in nitrites and then into phosphates. Then in fish-tanks the live plants feed on those elements and maintain low levels of them. In turtle tanks this is a problem. Because even if you have good filter with all the needed bacteria and working nitrate cycle, there's no one to eat the nitrates, nitrites and phosphates.
In your case - You have working cycle and that's why you can't detect any nitrates with your test, because they are turned into phosphates.
In fish-tanks, as I already said we calculate the quantity of light per liter. And most important we use flourescent (is that the word) light. Because it's cold light. Warm light which is used for turtle aquariums helps algae growth because of it's wide specter. I suppose the algae is very near of your light source

(personal experience) My room looks directly North. And there's no direct sunlight through the window at all. I had a fish-tank directly by that window and algae grew incredibly thick in 3 days. So I suggest reevaluation of terms - direct sunlight is very bad but direct daylight is not good either!
So after you've read all that wise-*ss crap I wrote... let's see what can you do.
In your case:
1. You can try changing one fifth of your tank's water every 48 hours for at least 2 weeks. This is done in order to minimize quantities of nitrates, nitrites and phosphates in the water.
2. You can't plant live plants, so the easiest way to fight algae is not for you.
3. Light! Try to direct your light source only at the basking area so few light would hit water. And most important think about the daylight in your basement is it heavily over your tank? Place the tank on the corner with most shadows
4. I guess you won't like this opportunity but I'll say it anyway. Siamese algae eaters (SAE) grow big. And an old SAE's maximum length is 15 cm (6") which makes them hard to get eaten by the turtle.
Good luck!
...you cannot KILL what you did not CREATE...