Metaphor and Simile
You will notice exactly how strict the distinction between metaphor and simile is by looking at these two examples: "Now I am a lake" (line 10) and "Rises toward her day after day, like a terrible fish" (line 18). The first example is a metaphor: The mirror compares itself to a lake. Although the comparison seems explicit, the use of the verb "to be" -- here, "am" -- actually obscures the comparison rather than revealing it. For example, if you were to take the statement "the mirror is a lake" literally, there is no comparison made, only an equivalence. The second example is a comparison between the reflection of a woman and a fish. This example uses the word "like" and therefor is a simile. Similes use the words "like," "as," "than" or "resembles" to make comparisons and may also be embedded in a sentence without those modifiers.
Personification as Metaphor
Even though the poem contains multiple metaphors, the metaphor involving the mirror dominates throughout. However, this is not your typical metaphor because it might be better described as a personification: Although the mirror is being likened -- that is, compared -- to a human, it is more significant that the mirror is given human attributes: The mirror is truthful, mediates, observes, and is faithful. You could also, for example, make the same argument for the candles and moon, which are compared to liars (line 12).
Some metaphors are more difficult to detect than others, especially when they do not involve a comparison between two nouns. For instance, the use of the verb "swallow" (line 2), although it is a human attribute, does not constitute personification. The mirror is not swallowing like a person does but is rather observing and taking in information. This act of gluttonous perception -- the mirror swallows everything "immediately" -- is compared to the act of eating with relish. So although the verb "eating" is not directly stated, the metaphor implicitly makes this comparison.
Technically speaking, the phrase "Now I am a lake" (line 10) can be read as a metaphor in which the mirror compares itself to a lake. However, it is possible that this statement is meant to be taken literally. After all, in this poem a mirror is able to talk, so why wouldn't it be able to turn into something else? And indeed the indecision shown by the mirror in the phrase "the candles or the moon" (line 12) is evidence that this may indeed be the case. Furthermore, the woman "rewards" (line 14) the mirror-lake with tears. The word choice of "rewards" makes sense only in the context of the speaker being a lake -- the reward is that the lake grows with the addition of tears. Whether or not you take this phrase "I am a lake" to be metaphor or reality will have multiple implications for how you read the rest of the poem as well as how you interpret the poem as a whole.