Becky has now become even smarter and better at faking niceness than before. She uses her new skills to worm her way into the Crawley family.
She endears herself to the two girls in her charge by not making them learn anything and generally leaving them alone to do whatever they want to do.
She then gets on prissy Mr. Crawley's good side by telling him how he smart he is.
Finally, she ingratiates herself with Sir Pitt in two ways. One, she's really pretty, and he's into hot young women. And two, she actually is really smart and starts to help him manage the Queen's Crawley estate and deal with all of his lawsuits in an organized way.
In summary? Becky is awesome and is making all kinds of lemonade with the Crawley lemons.
Oh, and here's some more info about Miss Crawley, Sir Pitt's rich old sister. She really loves her nephew. Not Mr. Crawley, but his younger brother, Rawdon. He's a dragoon (a kind of soldier), and an all around super-macho guy. He likes pool, cards, drinking, and the ladies. He got kicked out of college for this lifestyle and joined the army. Unlike his brother, he has certainly never been called "Miss Crawley."