"I have nothing in the kitchen."
Translation:Non ho niente in cucina.
That's not a valid sentence in Italian; you can avoid the double negative only if you put "niente" before the verb, as "niente ho in cucina", or in an interrogation, as "hai niente in cucina?", but the first word order is very rare (mostly poetry and a couple of dialects). "In la" is wrong as well, it's either "in" or "nella".
Nulla is usually a perfect synonym for niente; nullo is mostly an adjective meaning "equal to zero" in math or "void" in legal jargon (other meanings are archaic).
P.S. Back then I forgot to mention that "hai niente...?" doesn't mean "do you have nothing?" but "do you have anything?", so it's basically equivalent to "hai qualcosa...?"