How should we interpret the tone of Kate’s final speech in the play? Is she sincere? Sarcastic? Beaten down?

Courtship and marriage is the butt of jokes, games, disguises, innuendoes. In short, marriage is theater in The Taming of the Shrew. Just so, Katharina's speech, in the context of the play that precedes it, is deeply ironic. It would be one thing if, after subjecting her to such a cruel battery of taming techniques, Petruchio made the speech; but the fact that Katharina is given the last word - and also the longest speech in the play - is itself enough to raise an eyebrow. Perhaps Kate's speech is her way of putting on yet another act, of wryly offering one more illusion. According to this reading, Kate's subjection is a form of grand sarcasm, as she pretends to genuflect before the childish men who have spent so much of the play in comic confusion. This is how you play their game, she seems to say between the lines, and this is how you beat them at it.