I absolutely adore
Lila (pronounced LY-luh) and
Lily, but I'm not sure how I feel about the "Lilla" pronounciation. It intrigues me, but it sort of comes across as some sort of weird hybrid - like the person just couldn't choose. How about
Liliana? You could throw around quite a few nicknames with that one, even Lilla on occassion.
What about
Laila (LAY-luh, although it can be pronounced like LY-luh) or
Leila (LEE-luh)?