Let's eliminate a few possibilities. Your nitrite is not high enough to massacre any fish. Neither is your nitriate. Also, with your nitrates up to 20ppm, that tells me your cycle is complete.
Check your ammonia. If it's too high, it can be deadly. With your nitrites at 1 and your nitrates at 20, I expect your ammonia is 0. Confirm that.
What is the temperature of your tank? What fish did you put in? If your tank is at 80ish, it could feasably kill feeder goldfish or rosies. They keep the tanks at 70ish at the lfs. I've put two dozen feeders in my tank before to end up with half of them dead by the next day. (in a breeder basket to isolate them). The tank was at 81.
The chlorine could be the culprate. It's basically just bleach. It's a small amount but it's still toxic to fish. I would add a filter to mine even if the fish didn't die.
Get some Seachem Prime. Turn off the drip. Add 4 capfuls of Prime and put some more test fish in there. That should neutralize any tap water issues. See if that works. If so, I'd blame the chlorine.