Hard to tell what went wrong, but nitrate poisoning isn't the only risk.
There's also nitrite poisoning for example (related but separate), as well as various other chemicals that might have gotten into the water.
But the main risk with too many lights, especially for a small tank like that, is temperature. Lamps emit not just light but heat, and a small tank has trouble enough shedding excess heat as is, place bright lights overhead and that can lead to overheating unless you have an active cooling system in place (which most people don't, they're bulky and expensive, and can be noisy).
Even without the heat though, those bright lights may cause unwanted algal growth, algae that can (depending on species) leach toxic chemicals into the water (but you'd probably have noticed it if blue or black algae started growing).