Ok, so you used some existing sand with an existing bio filter, then threw in a crap ton of new sand, which proved to be loaded with more silicates than your biological filter could handle. At this point, it would have been ideal to let the tank catch up. However, patience may be a virtue, but it ain't easy, so you also added some livestock. This essentially added insult to injury to your bio filter.
The best thing you can do at this point is to help it out by removing as much of the gunk as possible. The water cleared because all your cloudy goodness settled. A couple gentle gravel cleanings would help get some of that out. Odds are, you will stir up more gunk and cloud your water again, so have some fine filter material around to help pull out more sediment. Repeat regularly until you see the diatoms diminish.
You also mentioned having a couple "dead zones" where the diatoms are strongest. Adding flow to those areas has already been mentioned, and it is sound advice. These areas will also collect extra food and whatever other stuff sinks in water. Not good, so try to find a flow pattern that minimizes the dead zones on a permanent basis. You may want to look into stronger pumps for the nano since the stock ones suck.
Rinse your food to remove phosphates, etc (a good long-term practice as well), and keep plenty of sand sifters in the tank with the amount of sand you've added. Keep it clean, especially since the nanos don't have great filtration. A mix of nassarius, cerith, and nerite snails is almost a must, and others can be added too. Be careful with cucs early in a tank's life - they are dirty (and possibly toxic, depending on the type) when they die. I'd also watch a conch in that small of a tank. I've had a 28 nano for years and they never make it long, probably due to starvation. A note on pods - you probably got some in with your chaeto and existing sand if you got it from a healthy reef tank. If not, seed some.
Regarding testing, keep in mind that it is often inaccurate. You are testing what is in the water column, which doesn't account for whatever nuisance algae is using up. If you have bad stuff, something is feeding it.
For a light cycle, you can cut to 6-8 hours if you want, but the nutrient issues are still the main issue, not the photosynthesis. Don't light it up all day long or anything, but I'd keep my focus on nutrients if I were you.
For now, get a small cleanup crew, and be patient - it will cycle out. Keep up the husbandry (water changes, light sand cleaning, and fresh filter material), too. The nano will require that forever, but particularly in the early stages.