"Whereas the Viennese instrument was light and delicate, the English pianoforte, developed chiefly by the Scottish pianoforte-maker John Broadwood, was noted for its brilliance and power. Broadwood...made several modifications to the pianoforte: adding more strings per note and iron braces to the wooden frame and introducing a sustaining pedal, instead of a hand-controlled stop.

Broadwood also worked closely with pianist-composers, and in particular with the Bohemian Jan Ladislav Dussek. ... Dussek's collaboration with Broadwood resulted in extending the compass of the pianoforte keyboard from five to five-and-a-half octaves in 1791, and to six octaves in 1794."
From chapter seven of "Musical Visitors to Britain" by David Gordon and Peter Gordon

Dussek was a composer/virtuoso member of The London Pianoforte School, founded by Clementi.
From the same source:
"The London Pianoforte School, which consisted almost entirely of foreign visitors like Dussek, had an identifiable musical character, reliant on dramatic effects, song-like melodies and keyboard virtuosity. Much of the output of the school is dominated by technical brilliance, and J.B. Cramer, a German composer, who was taught by Clementi, developed the genre of the study, or Studio, as he called it."

When Frank Churchill asks Emma if she has heard of Cramer, I suspect him of having a private joke at her expense, knowing more about music than he pretends- enough to know full well that she has never heard of Cramer.
The Irish Melodies that came with the Cramer, might have been the third volume of Thomas Moore's Irish Melodies (although, in February 1814 it was Moore's satire lampooning the Prince Regent, "Letters Intercepted or the Two-Penny Post" that was heating the press of the London newspapers, or at least, 'the Courier' newspaper).

Clementi also developed a new method of teaching this new style of playing this new instrument, with a progressive syllabus. When he went on tour to Europe, he collected pieces from contemporary composers as well as the old masters, sometimes commissioning pieces especially for his tutors for music students, as well taking pieces to be published in his London sheet music business.

It is a possible that Jane Fairfax learnt to play in by the syllabus of the London Pianoforte School -starting with Clementi's "Introduction to the Art of Playing on the Pianoforte"(first published in 1801), methodically working through his volumes on The Selection of Practical Harmony (published 1801, 1802,1811 and 1814). The dates these tutors were published are timed just right for her education, with her going to the Campbells in 1802 and continuing with her piano master in London until 1812.

These were superior tutors, exposing the student to the best of the old and modern composers from the very start. Poor Miss Taylor had learnt how to teach before she arrived at the Woodhouses in 1797, so she was probably had to adapt the ideas of someone like C.P.E Bach, who taught for the harpsichord, to the new instrument for the Miss Woodhouses.