Indirect statement 'I know thee that thou art' ?

For an indirect question Greek often says something like οἰδα σε ὁστις εἰ… (‘I know thee who thou art’)

By analogy, is it possible to say οἰδα νιν ὁτι ἐστιν… for ‘I know that he is…’?

Sure. Though νιν is Doric. From the LSJ article on οἶδα: 5. c. acc. folld. by ὡς, ὅτι, etc., οἶδα κἀμαυτὴν ὅτι ἀλγῶ S.El.332; ἐάν τινα εἰδῶσιν ὅτι ἄδικός ἐστι Pl.Prt.323b, etc.

My translations for the examples (possibly with errors, as always):

οἶδα κἀμαυτὴν ὅτι ἀλγῶ - “and I know myself that I hurt”
ἐάν τινα εἰδῶσιν ὅτι ἄδικός ἐστι - “if they should know someone that he is unjust”

To add to jeidsath’s response, this construction is generally known as “prolepsis.” The Cambridge Grammar of Classical Greek (sec. 60.37, p. 720) describes it as where “topics of subordinate clauses which follow the matrix clause are . . . syntactically integrated into that matrix clause.” The example given is Sophocles Philoctetes 549-50: ὡς δ’ἤκουσα τοὺς ναύτας ὅτι | σοὶ πάντες εἶεν συννεναυστοληκότες . . . "But when I heard that the sailors all belonged to your crew . . . "

Smyth also discusses prolepsis in sec. 2182, with further examples:

http://www.perseus.tufts.edu/hopper/text?doc=Smyth+grammar+2182&fromdoc=Perseus%3Atext%3A1999.04.0007