Im sorry to say, but you really can't keep any fish in a tank that small. The water levels will fluctuate too frequently with a fishes high bioload, and the stress will eventually kill them. This is probably why your set up didnt work before.
If youd like to keep your neon tetras, Id recommend moving them to at the very least a ten gallon tank. Those little tiny tanks really can only hold invertebrates like small shrimp or snails. Maybe one or two small male guppies, if you're very diligent on your upkeep. As for the other fish, Im not at all sure what kind it is, but it probably needs at least ten gallons, also.
You should look up how to cycle your tank. Basically, it just means running it with a filter, light (maybe plants or whatever) and some source of ammonia. Fish food or even a small actual fish (like one of those male guppied I mentioned) can provide some ammonia for the growing bacteria population that you'll have going in your tank. Hardy fish like a guppy can survive the "cycling" of brand new tanks. Neon tetras tend to be more delicate and won't survive the first few weeks of a new tanks life.
In reality, you should get a ten gallon (at the very least). Set it up with a filter, heater, light, and some sort of gravel or sand. Add a tiny pinch of fish food every couple days, or add a single male guppy and keep him lightly fed (you can always remove him later, this is just to get your tank established). Wait a few weeks to a little over a month and test your water during this time to make sure your tanks cycle stays on track. After the cycle is complete, start slowly adding fish. Maybe a couple every week or so. Change a portion of the water weekly. Dont overfeed the fishes.
This is all just based on my own experiences and the research that Ive done to better my understanding of fishtanks. I now have several tanks, and none of my fish have died in months!