Your question leads us to consider Poe's purpose in creating this poem with its elusive references to the beautiful and lost Lenore. We know that the speaker loved Lenore, and we know that everything surrounding him reminds him of her--even the velvet cushion of the chair he finds himself reclining on:
This and more I sat divining, with my head at ease reclining On the cushion’s velvet lining that the lamp-light gloated o’er, But whose velvet-violet lining with the lamp-light gloating o’er, She shall press, ah, nevermore!
So why don't we know more than that? It may well be that too much specific information about Lenore, such as her cause of death, would draw away from the mysterious quality of the poem with its relentless emphasis on loneliness and loss. What matters to the speaker is not how Lenore died or who she was, but the fact that she is gone, forever. The haunting repetition of "Nevermore" at the end of each stanza continually pounds the meaning and permanence of the loss into the speaker's mind.
Keep in mind, too, that the speaker's state of mind, not our state of awareness, is what is important here. The poem is written in the first person point-of-view. Since the speaker's agony is centered on Lenore's loss, it really wouldn't make sense for him to tell us who she was or how she died. We don't need to know the specifics of why or how she was important to him; we know very well that she meant everything to him. Without her life is hollow and empty.
We--the ones who might want to know more about Lenore--aren't even there to the speaker. It is as if we are eavesdropping on his private, solitary grief. With the unrelenting, persistent rhythm of the poem, the poetic sound devices (such as alliteration, assonance, consonance, rhyme, and onomatopoiea), and that famous refrain ("Nevermore!"), Poe makes us FEEL the speaker's grief. We know how much Lenore meant to him and we accept his refusal to believe he can ever be reunited with her. Furthermore, he knows he will never be able to forget her, that he will never be at peace.
In another way, the lack of specific information about Lenore underscores not just the mysterious, haunting tone and mood of the poem but HER mystery as well. What a special person she must have been to elicit such grief! She could be the special love of anyone's life; leaving her identity a mystery makes her a more universal figure of love.