Some tap water comes from the tap with nitrate and/or nitrite in it. I would check your water straight from the tap with the test kits first to see if it's there or not. I think it is possible to see nitrite after 10 days but probably not nitrate. Nitrate is more commonly found in tap water especially if it gets agricultural runoff. It's also possible the test kit is faulty. Most test kits are not very good. Did the test use dry pills, liquid, or dip sticks. I'm a chemist and can tell you that some are very hard to read and some are can go bad if it's not new. For some water chemistry information and tank cycling, see
http://www.fishpondinfo.com/chem.htmI'm blessed with soft, moderate pH water which danios like. You can continue to mix distilled with tap water or spring water with tap water. I don't think you have to go to 100% spring water (100% distilled will kill almost all fish as it's devoid of any anions or cations; sorry, I'm a chemist). If your water is that bad, a reverse osmosis system which is very expensive might actually save you money over time than buying bottled water. But, a 10 gallon is not that big so probably not. I would stay with water changes in the 20-25% range if you're changing with water of different chemistry than what's in there. You don't want to change pH or hardness too quickly. I usually say to try to get fish that like the water you have from the tap instead of trying to fix it. There are lots of hard water loving fish out there. Danios aren't one of them though. Until things are under control, I wouldn't get any more fish. You'll find that each of us aquarists has our own ideas and sometimes all of them are right and work! Hope we didn't confuse you too much.