Jane Fairfax was actually going to Hogwarts when she moved out of Highbury as a child.
She left before she was 9, the reason being that she was going to be taken care of by the Campbells after her parents died. That's just the right time for her to go to Hogwarts, which would have been in operation for a few centuries at that time. It would have been about 3 years after her graduation by the time she returned more permanently at the age of 21 during the novel.
The Campbells probably have one magical person in their family who recognised that Jane was showing signs of magical ability during her visits. By inviting Jane to live with them, Colonel Campbell was not only taking care of his late friend's daughter, he was also ensuring that there would be a magical adult to teach her how to control her magic until it was time for her to go to Hogwarts.
Mr. Knightley, tragically, died young.
Emma married again. Her second husband's name was Norris.
Frank Churchill killed his aunt.
Okay, so it's not that likely in a Jane Austen novel, but I've seen this one floated a few times and there's no denying that the timing of her death is very convenient, coming just after Jane has broken the engagement and just before she goes off to be a governess.
If Mrs. Churchill really would disinherit Frank for getting engaged to a poor woman, he would have two options to salvage his relationship with Jane. One would be to declare it openly and be chucked out of Enscombe and the will. His father and stepmother would certainly take him in, but he'd be forced to enter some kind of profession so that he could support himself and Jane. Given his Mr. Weston's business connections and the large number of sympathetic neighbors, this would be completely doable, but it would delay the marriage and prevent Frank from becoming a properly idle gentleman until he'd saved up as much as his father had. Or... he could prevent Mrs. Churchill from ever objecting to his activities again, keep his comfortable lifestyle, and marry immediately.
- In support of this, it's noted that Mrs. Churchill was taking strong drugs for the chronic pain of her illness, so it would have been very easy for Frank to "accidentally" give her an overdose, or even just refrain from preventing her from overdosing herself. And since she was dying and in chronic pain, Frank might well have been able to persuade his conscience that giving her a quick and painless death was as much mercy as murder.
