"The Hungarian girl lives in this house."
Translation:A magyar lány ebben a házban lakik.
There's no reason for it to be definite. You only use definite conjugation if the verb has an (implied or explicit) definite direct object. "Lakik" here doesn't have any kind of object.
There would be one way to say this thing with a direct object, similarly to "The Hungarian girl occupies this house":
"A magyar lány lakJA ezt a házAT".
So, there you go, it is definite. But this is a different sentence.
I don't think ezben is a word. ebben means in this, while ennel means by/near/at this. Keep in mind that these do not correspond perfectly to the english words.