Is easy to prove that this is not a UFD, since $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ where $2, 3, (1+\sqrt{-5}), (1-\sqrt{-5})$ are irreducible and are not associate to each other.
I'm not sure how to prove that every $a \in \mathbb{Z}[\sqrt{-5}]$ has a factorization. The best thing I could think of is that, since every number has norm greater or equal than $1$, and $|ab| \geq |a|$ for every $a$, then the process should end at some time.
Your observation, the fact that units are precisely the elements with norm 1, and that the ring is Noetherian is enough to conclude it is a factorization domain.