We do not know exactly how Annabel Lee died in this poem.
We are told two things that might be relevant. First, we are told that a cold wind blew from a cloud and that was (it is implied) what killed Annabel Lee. So from that, you could imagine that she might have died of pneumonia or of exposure (freezing to death).
However, if you look a little earlier in the poem, you see that it was angels from heaven ("winged seraphs") that caused Annabel Lee to die. They were jealous of how perfect the love of Annabel Lee and the speaker was and so they killed her.
So you can pick which of these you want to use for the answer. You can say the angels killed her or you can say she died because of the effects of the cold wind.