Emma is wrong in not paying proper attentions to Miss Fairfax and the Bateses. That is certain.

Emma does not ask for Jane Fairfax for gossip: she asks for impressions. Since Author Austen has her gentlemen and ladies frequently do the same in her novels (i.e., P&P chs. 14, 26, 44; S&S ch. 26; P, chs. 6, 11, 14, 15, 17; MP, ch. 29), I doubt that it is impertinent for Emma, a gentlewoman who recognizes and despises vulgarity, to do so. I can not assume Emma's questions were 'over-probing,' when Mr. Knightley, who is there observing a pleasant change in Emma towards Miss Fairfax, sees nothing of the sort. Emma even hides her resentment so well, that Mr. Knightley is surprised that there was any, when he visits the following day.

Author Austen does not cover every subject spoken of that evening, when Emma thinks it over: only the ones on which Emma bases her return to original dislike. Therefore, it may seem as if Emma did nothing but carp on Mr. Dixon and Mr. Frank Churchill, while Jane Fairfax politely kept her own counsel in the face of such impertinence. But that can not have been the case, or Mr. Knightley would have noticed it, and reprimanded Emma about it.

You may say Mr. Knightley is not 'forcing' Emma and Miss Fairfax to be friends, but whenever Emma honestly points out Jane Fairfax's faults, he is disappointed, and whenever she repeats admiration or sympathy for her, he is more than pleased. His comments, and the tone I hear behind them, seem to imply a great wish of their being intimate: not merely Emma being just to Miss Fairfax, but actually finding a friend in her. Part of that might be his hope that Emma would then be less with Harriet Smith, but I think it undeniable:

"I always told you [Miss Fairfax] was [reserved] -- a little; but you will soon overcome all that part of her reserve which ought to be overcome, all that has its foundation in diffidence."

Emma is to overcome another's reserve -- that does not sound like inviting her by to play some evenings, and making visits of course. He refuses to acknowledge Jane Fairfax's overdone reserve, but, even had he been right (and events will prove him wrong), if Emma sees it, how could she enjoy the other's company?

At his short statement of disappointment in her, Emma appeases him, and

Mr. Knightley looked as if he were more gratified than he cared to express;