To me, the turn of this sonnet begins at the line that starts "Yet in these thoughts..." By saying "yet," the speaker is clearly saying that what came before is going to be different from what will come next.
The speaker's tone after the turn is completely different. Before the turn, he was being all pathetic. He was talking about how much he hates his life and how cursed he is.
But then, once he thinks of his love, he gets really happy and confident. Then, he wouldn't even change places with a king.
While Sonnet XXIX has the rhyme scheme of a Shakespearean sonnet, its thematic format is that of the Petrarchan sonnet in which the octave states the problem: the brooding poet senses his misfortune as he curses his fate and wishes he could possess the artistic talents and friends that others do, all of which would greatly lift his spirits,
When in disgrace with fortune and men's eyes,
I all alone beweep my outcast state....
Wishing me like to one more rich in hope--
Then, the sestet provides a solution to his dark broodings when he reflects on the fortunate aspects of his life as he "haply" dwells on the love bestowed upon him and his spirits lift,
Haply, I think on thee,--and then my state...
...sings hymn's at heaven's gate.
It is here at the beginning of the sestet, then, that the "turn" occurs. With the love of the woman, the speaker is relieved of his isolation and is no longer alone; now there is meaning in his life, and thus fortified by love, he can again be optimistic.
All the lines end with commas up to the line before the turn,
With what I most enjoy contented least;
which ends with a semicolon, indicating a longer pause in preparation for a new thought in a new tone, almost as if his sole consolation had just occurred to the speaker. Then, as noted in the above answers, the poet abandons his litany of dejection and begins to approach the only image in the entire sonnet:
Yet in these thoughts myself almost despising,
Haply I think on thee, and then my state,
Like to the lark at break of day arising
From sullen earth, sings hymns at heavens gate.
The image of the lark soaring into the sky while singing is one of the most beautiful things in Shakespeare. At least two things make it especially effective. One is the contrast with the morbid thoughts which had gone before. The other is the alliteration of "S" sounds in "sings hymns at heaven's gate." There are actually four "S" sounds in these words because the "s" at the end of the word "sings" blends into the word "hymns," and there is an "s" at the end of "heaven's." The adjective "sullen" was probably chosen to add yet another "S" sound to this wonderful line.
This sonnet is different from some. Many sonnets have their turning point in line 13. This one is in line 9, "Yet in these thoughts..." That word "yet" is a big clue, and the transformation is not subtle, but complete. In the first 8 lines the speaker almost hates himself - he tells us this in line 9 "myself almost despising" - but then his heart takes flight like a bird. He feels so happy and free, he could almost fly. His tone goes from sullen to joyous simply by thinking of the woman he loves.