Sounds like the tank is too small, zebra danios really need at least 20-gallons (and they also do better in groups). It might sound weird getting more danios when yours is a bully BUT that actually helps sometimes! The aggression gets stretched out within the group of danios--plus he has some fish of the same species to interact with (therefore distracting him). Also, definitely get more neons! They absolutely need to be in schools (get at least six).
The other problem is that leopard danios are a bit more timid and fragile compared to zebra danios. You could try getting more leopard danios, too, so you have some strength in numbers. Make sure you have plenty of hiding places and plants as well so they can get away if needed
If it was me, this is what I'd do:
-Buy a 20-gallon aquarium, a few more zebra danios, a few more leopard danios, and a small school of neon tetras (I'm not a fan of mixing neons with danios like this but you already have a neon so you might as well get a school to keep him healthy).
-Quarantine the new fish (you probably know this but I'm just throwing it out there, pet store fish can sometimes be diseased). Meanwhile, set up the 20-gallon and get it cycled and stable (you can cycle quickly by using gravel from an established tank).
-Make sure there are a lot of hiding spots and plants (real plants are always better but fake works too). The more hiding spots, the less territiorial aggression
-Add ALL of the fish to the 20-gallon at the same time. Don't add in one group before another (and if you have to, add the new guys first). You don't want your zebra danio to establish territory early and have an advantage (and a reason for aggression).
While this is going on, get a breeder trap and separate the zebra danio from your other fish. Feed the other fish healthy, filling foods (frozen and live foods, in other words) to get them back in shape