I see... The cories are very sensitive to changes in water parameters... I am assuming the cardinal tetras are too (neons are). Not sure about guppies but I don't think they are recommended for cycling the tank either..
Do you have anybody who could take your fish in while you finish cycling? You could just get some other fish and cycle with them. I'd try the hardier tetras (I have red eye tetras and they've done great... serpae tetras might also be hardy enough) or zebra danios. I cycled my tank with 4 red eye tetras and it's cca 33 gallons. You have to have enough fish to spike the amonia enought for the bacteria to kick in but not too many that they produce too much amonia and die.
I read that there has to be spike in amonia and then nitrites to trigger the bacteria growth. I'm guessing by trying to keep the levels low so that it's safe for the fish there was never enough of a spike for a colony of bacteria to establish itself. I know you don't remove bacteria when doing water changes but I would guess that removing too much of their food would also dampen the growth.
I hope you find a solution and your fish make it!