If you follow the link on this site, it says that Lila comes from the Indian name Leela which I assume is pronounced LEE-la.
However, I have known more than one Lila and have encountered several characters in books (audio) and films named Lila. Every single one I have encountered pronounced it LYE-la. I assume it is because of the pronunciation of the name Delilah, especially when Lilah is spelled with the H. Being from America there is no way that I would look at the name Lila and think it was pronounced LEE-la. It is Lila to me, all the way. :o)