Let us analyze the given information:
- From condition I: Deepa’s mother is the sister of Laxmi’s father. This means Deepa’s mother and Laxmi’s father are siblings. Therefore, Deepa’s mother is Laxmi’s aunt.
- From condition II: Deepa is the daughter of Laxmi’s grandfather’s only son. This means Deepa’s father is Laxmi’s uncle.
So, Deepa is the niece of Laxmi.
Thus, the correct answer is \( \boxed{1} \).