My name is Danica. It is pronounced Da-nik-a. I have seen some discussion on the pronunciation of it and I wanted to add my two cents. I rarely get someone who can appropriately pronounce my name unless they are European. Most Americans pronounce it Dan-eeka.
I've heard "Danica" pronounced two different ways: "dan-ika" and "daw-nika." It's a very pretty name; I like it a lot. It is also the first name of the main character of Hawksong by Amelia Atwater-Rhodes.
My name's Danica. I live in the UK, and I pronounce it DAN-i-kah. (Though I wouldn't say there was obvious emphasis on any of the syllables really) There seems to be a great range of differing pronunciations for this name.
I derived this username, Weaponstar, from the meaning "morning star", because a morning star is also a medieval weapon (a bit like a mace).