The speaker of "To Melancholy" is addressing—you guessed it—her own melancholy. Imagine that you could open up your own brain, pull out your bad mood, set it on the desk in front of you, and talk to it. That's what the speaker of this poem is doing. (Metaphorically speaking, of course. she's not literally lobotomizing herself. That would be gross.)
But the speaker actually seems to like her melancholy mood. "Melancholy" doesn't just mean "sadness" in the context of this poem (and according to its definition in 1785). It has to do with being thoughtful and introspective and thinking deep, sad thoughts. It's the kind of mood that inspires emo rockers to write the kind of song that we all listen to after bad break-ups.
And that's the kind of mood that the speaker is addressing in "To Melancholy." She actually enjoys the feeling: it gives her a chance to brood and to think her deep, sad thoughts. She finds her melancholy to be soothing (14), the same way that we all enjoy listening to the same sad and melancholy song over and over again when we're feeling down in the dumps. She's describing that so-bad-it's-good feeling that almost everyone has experienced at least once in their lives.