of ones faults is already half way to improving.

Depending on what one might consider Emma's faults, I have made these following observations;

-Match making itself is not being considered a fault, unless Mr and Mrs Weston share this fault. While the Westons only hoped for the match in their head, Emma was forcing it.
But she has learned something at least from this;
CH:40
"I give you this caution now, because I shall never speak to you again on the subject. I am determined against all interference. "

-narrating her fantasy about Jane Fairfax and Mr Dixon to Frank Churchill was wrong.
Though she won't give up her fantasy, she's ashamed of telling it to Frank Churchill.

-her opinion about the people from the lower class has undergone a change (half way as I mentioned).
That she attended the Coles' party, even if it was for the pleasure of being with Frank Churchill, is a step forward.

She has 'ultimately' been influenced by the fact that the Westons and Mr Knightley are going so it cannot be wrong, though earlier she had no hope, from them.
It can be said that its because she wants to go she's comforting herself with these things, but nevertheless it is a compromise which shows hope.
I doubt she would have gone only for the pleasure of being with FC if Mr Knightley and the Westons were not going.

This step may lead to socializing more, and changing her attitude. It's this non socializing (and I don't mean visits to Miss Bates) which is the root cause of her narrow view IMO

-a change towards Miss Bates.
CH:43
>Time did not compose her. As she reflected more, she seemed but to feel it more. She never had been so depressed.[.......] and Emma felt the tears running down her cheeks almost all the way home, without being at any trouble to check them, extraordinary as they were.

So her reaction here is extraordinary, and I see a complete reform regarding Miss Bates.

I'm not inclined to see anymore faults, so I'm quite satisfied with her progress.