NO2 at 5 or higher is a REAL problem. Not just for fish, but for people, too. Double and triple check this: Bad test? Where is the NO2 coming from? Maybe check the neighbor's water, too.
Look up methemoglobinemia for more info.
RO water is a great place to start. No minerals, salts or anything else. Then add what you want. Just the right levels of carbonates, calcium, magnesium and so on. I use baking soda or potassium bicarbonate for KH and Seachem Equilibrium for GH.
Spring water may vary from one batch to the next, I would not trust it.
nitrite 5.0 or higher
nitrate < 5.0
dKH 13? (chart ends at 12 drops, it took 13)
GH greenish from first drop, never orange, are drops to degrees a 1:1 proportion?
(all API kits)
pH: ought to correct itself when the other things are corrected.
NO2: This is bad. Maybe the test kit is bad? Check this with another test kit, even if you have to take a sample to a store. Do not drink this water.
NO3: 'under 5' is OK.
KH 13 German degrees of hardness: This is holding the pH up there. Reverse osmosis would remove the carbonates. Yes, it is oK to call it 13, just because the paper stops its chart at 12 does not mean the kit is not good for higher values. Both GH and KH are 1:1. One drop = One German degree of hardness.
GH 0-1 degree: This is not good. Even fish that thrive in the softest water need some calcium and magnesium. Plants need them, too. However, low GH makes it good to run the water through a reverse osmosis system. The RO membrane tends to clog when there is too much calcium.
There are portable RO units: hook it up to a sink when you want to use it, then dry it off for storage. This will also get rid of the NO2 in the water.
How I would handle this:
Use RO water.
No tap water until the NO2 issue is resolved. (Bottled water for people)
Add minerals to the RO water to make the GH and KH suit the fish.
If it is a black water species, filter the water through peat moss to add the organic acids these fish like.