I am sorry to hear that you are having troubles with your livestock! Having said that, you did move ahead kinda quick. And from my understanding have a lot of issues to solve. One problem that you will have is that you have a small amount of water than most salt tanks ( hence the nano). Anyway, any changes in the water quality will happen much faster for you then someone with 30 or 55 gallons of water.
Example: if your die off in the tank creates 10 mg of NH3 into 49.2 L (13 gallons) of water you will have a concentration of 0.20 mg/L which can be noted as a small but alarming spike. A water change would most likely fix this for you.
As for someone with a 208.2 L (55 gallon) the concentration would be 0.048 mg/L and would not even show up on 95% of the test kits out there unless they are digital.
Just so you know ppm and mg/L are so close that they can be more then safely interchanged.
Anyway, please post all of your water parameters for us to read. Run the same tests on the distilled water and post them if there is something wrong. Distilled is not causing your problems unless there is something added to the water that should not be there. The problem is a juvenile tank that is not cycled and has a poor biofilter. Take a step back, work slowly, and fix one problem at a time so you know what works.
As of now I would remove all life from that tank because your NH3 levels are going to stress everything to death. As they die and after they die they are going to keep releasing NH3 and your not going to see any changes, only more death in your tank. I hate to sound so dreary but suffer the loss now and get things stabilized. Try asking your LFS or someone that works there if they will host your livestock for a month until you get things fixed. Or set up a hospital or Q tank and do this yourself.
Once you get the livestock out. Do a 90% + water change leaving your LS and LR in the tank and let it sit. If you would like to add anything, you could add Cycle, Stability, or Prime. Keep it simple and slow. Don't put anything back in that tank until NH3, NO2, and NO3 all read zero. NO3 can be a MAX of 20 ppm.
Here is a picture of one of my diatom blooms while cycling.
Yuk! I cleaned the gravel and it came right back. I even had green hair algae growing as well. Once water parameters became good I added my CUC and it was gone in four days flat.
I hope this helps! Sorry to sound the like the downer. But what you have is normal except for the fact that it is harming your livestock and making the problem worse.