In "Annabel Lee," how does the speaker react to Annabel Lee's death?

The speaker's reaction is certainly one of undying dedication. 

"Annabel Lee" can be read in the context of American Romanticism. It employs language to evoke mood and emotion; it uses elements of the supernatural (e.g., angels, winged seraphs of heaven, and demons) to illustrate the extraordinary nature of the narrator's love for the subject.

Because you are focused on the narrator's reaction, I would look to the last three stanzas. In the fourth stanza, we learn how Annabel Lee died: "That the wind came out of the cloud, chilling / And killing my Annabel Lee." Exposure to cold and dampness (she lives by the sea) was a concern in the 19th century. Perhaps it was pneumonia that took her. 

In the fifth stanza, the narrator expresses what was different about his love for Annabel Lee. They were spiritually connected spiritually long before her death: 

But our love it was stronger by far than the love

Of those who were older than we—

Of many far wiser than we—

And neither the angels in heaven above

Nor the demons down under the sea,

Can ever dissever my soul from the soul

Of the beautiful Annabel Lee:—

In the first three lines, you have the voice of a youth who believes that no one can understand the depth of his emotion, another characteristic of Romantic literature. For him, their love transcended age, and notions of good and evil. His evocation of demons and angels also suggests the possibility that their love transcended the boundaries of religion and the church.

In the final stanza, the narrator describes how Annabel Lee is present in his memory:

For the moon never beams without bringing me dreams

Of the beautiful Annabel Lee;

And the stars never rise but I see the bright eyes

Of the beautiful Annabel Lee;

She has thus become a part of everything in death. She belongs not only to the narrator, but to the universe. 

The last few lines are less spiritual. The poem becomes dark, even morbid:

And so, all the nighttide, I lie down by the side

Of my darling, my darling, my life and my bride,

In her sepulcher there by the sea—

In her tomb by the side of the sea.

Here, it seems that the narrator wishes to join her in death: He lies down by his "bride" in "her sepulcher," in "her tomb."

Depending on how one wishes to read the poem, this could be affirmative to Annabel Lee's memory: the speaker will never forget her. Or it could be dark: he is trying to climb into the grave with her. The latter reading would make it more explicitly Gothic, and much of Edgar Allen Poe's work is in the early American Gothic tradition.

The narrator's reaction can thus depend on one's choice of how to read the poem.