This line shows that the mirror tries not to be biased when judging appearances, which, for people, is a hard thing to do. The mirror says that love and dislike don't cloud the way that it views people or objects. For most people, love and dislike definitely affect how we see the world and ourselves, as we see in the woman's distress with her own appearance later in the poem.
I have looked at it so long I think it is a part of my heart (7-8)
The mirror's connection to the opposite wall shows that when we look at something long enough, appearances can become more that just physical. We internalize what we see – in the world, and in the mirror.
Searching my reaches for what she really is (11)
Here, the woman is looking at her appearance in the water, and the depths beyond, to find something deeper in herself. As we discovered with the mirror's connection to the wall, appearances are significant to our identities. But this woman is not satisfied with what she sees; appearances aren't enough for a complete, satisfying answer to what she is searching for – herself.
"I am important to her" (15)
The lake is telling us in plain words that it is important to the woman. Of course, she could have some sentimental attachment to the lake in general, but in this poem, the reason why she goes to the lake is to look at her reflection. So, we can make the connection that if the lake is important, her appearance is important to her. Dissatisfaction with her appearance, then, would explain her distress earlier in the poem.
In me she has drowned a young girl, and in me an old woman Rises toward her day after day, like a terrible fish (17-18)
The woman's fears of aging, and the sorrow she feels over her lost youth, have become physical through the metaphor of drowning and rising in the lake. These two lines are scary – not only has a young girl drowned, but a scary fish is also rising from the depths of the lake. Through watching herself in these waters, the woman has seen what she used to look like, and be – a young girl – die, while what she is becoming – and old woman – is rising up to meet her. We all know, at some level, that we are becoming older, but by watching herself in the lake, the woman knows this on a visual level, made more vivid by the image of drowning and rising in the waters.