I think it needs the painting to work, and I don't think there's anything wrong with that. In fact, this poem is a marvellous take on the painting and it made me see and reflect on things I wouldn't have done otherwise. In the third stanza, the "it" definitely refers to the woman's eye. And in the painting one of her eyes is indeed visible; it's very tiny and I probably wouldn't have noticed it if the poem hadn't led my own eyes to it. The poet interprets the look in the woman's eye as "unsparing" (in the couplet). And as the poet implied earlier in S3, her (unsparing) gaze as she looks at herself in the mirror is akin to Lautrec's own way of gazing at things and seeing "unseen faults", faults that might not be visible to an ordinary onlooker.
Lots of layers in this poem, just as there are in a painting. I admire both the poem and Lautrec's painting.

Well, I seem to be somewhat at odds with others here, for it is the stanza others don't like, S3, that I think is the meat of the poem. Without that stanza and the couplet that resolves it, I wouldn't have much time for the opening of the poem which seems, on its own, a bit precious to me. I do agree that the syntax is screwed up, mostly because of the use of the word at.

His eye, so good at unseen faults in others
and at his own, picks out in her its kin,

To be good at unseen faults in others or good at his own faults doesn't really make sense. The lines could so easily be rearranged to be clearer.

His eye, which sees the unseen faults in others,
and in himself, picks out in her its kin

Nemo

I'm with those who say this one fails to take off for them. The first two stanzas are well and neatly-phrased. Nemo's suggestion vastly improves that quality in the third stanza. Overall I'm with David, Michael & Quincy, who find themselves underwhelmed by the whole, its competence notwithstanding.

What a great thread. I feel the critiques are spot on. The poem is not as ambitious as it might be and rather coldly analytic in its tone; yet it I do think it works on its own terms and is well-crafted overall.

The "problem line" invites an even easier fix:

His eye, so keen to unseen faults in others
and to his own . . .

Although this sonnet shows attention to detail and competent craftsmanship, it fails to move me, or even to google the painting. It needs another angle. As it sits now, it is merely description.

I like this very much. It's a fine companion piece to the painting, with which I was unfamiliar before reading the poem, but I think it stands on its own as well. It doesn't bother me that it's quietly descriptive, which seemed to be a problem for some readers.

Kate's suggestion regarding lines 9 and 10 seems a good one to consider, since it makes the meaning clearer at first glance.

Fine work.