I'm not sure this usage of fortnight was ever common. It's practically unheard of today, though we do still use the format with week. I'm not sure about modern Americans, but to most British native speakers...
"(I'll see you) Monday week"
...means I'll see you on Monday a week after the first Monday following today. By the same principle, Jane means a fortnight after "that day"*, but at the very least you'd raise a few eyebrows if you used the form today (many native speakers wouldn't just find it "odd"; they wouldn't understand you at all).
The sentence also sounds stilted because of the way it uses the word period. Jane is actually specifying the date on which she will assume the post of governess, rather than the "period" between the date of speaking and the date she starts work. In modern English she'd say ...as the date for assuming the post.