The Voice in the Corner spanking blog had a post on “the Markham Project.” “Miss Markham” seems to have been a retroactive creation, a shared name used in various femsub spanking writings by various hands over a considerable period of time. Some go as far back as the Englishwoman’s Domestic Magazine of the 1870s.
The Markham Project was apparently instigated by a professional governess and educationalist, Miss Elisabeth Markham between the 1880s and the early part of the 20th century.
Her aims were to encourage best practice in the training of governesses and the education of young women. She apparently advocated corporal punishment of young women over the age of 16 and unmarried ladies “up to a late age.” Her philosophy was to show that women were not frail creatures and that spanking, caning, birching and other “robust forms of punishment” were essential for discipline and the well-being of “females of all social conditions wishing to better themselves.”
I think a lot of spanking/flagellation erotica of the period imitated the debates over corporal punishment in school and domestic settings, both as protective imitation and as tongue-in-cheek parody. I very much doubt their ever were these flagellation-obsessed schools you read about.