The mirror does narrate this poem, and so it is personified in this way. However, when the mirror compares itself to a lake, saying, "I am a lake," this is example of metaphor. The mirror explains, in the final lines, "In me she has drowned a young girl, and in me an old woman / Rises toward her day after day, like a terrible fish." In other words, "she" -- the woman -- has watched her youth slowly disappear over the years of looking into this same mirror, and the mirror interprets that as a kind of "drowning" of the woman's youth.
Just as the woman's youth seems to fade away into the mirror, so does old age seem to be getting clearer and closer, rising toward her within the mirror's surface. The "terrible fish," then, is the specter of old age, the total loss of youth and vitality. This is likely why the woman looking into the mirror "rewards [it] with tears and an agitation of hands"; the woman watches herself age slowly, like a huge fish rising to the surface of the water, and she cries and wrings her hands. It is painful to grow old.