The current headmistress is Lady Catherine Loveacle, who has held the post for 10 years. Like all previous headmistresses, she is a female-line relative (in this case, a second cousin) of the Grand Duchess of Hallerton. Unlike some previous headmistresses, she has an extensive background in education, and had served as the principal of a public school before taking her current position.
Lady Loveacle began her career as a teacher of mathematics, and taught full-time even as she pursued an advanced degree in school administration. Her family name opened a lot of doors, but also put a few obligations on her. Still, she enjoys her profession.
Originally, the Royal Preparatory Academy (the aforementioned most prestigious school) only educated males, and only members of the aristocracy. The Hallerton Girls' Academy, as it was first known, was created to be the female counterpart. When Royal Prep started admitting girls (and non-nobles,) the HGA rebranded itself as the HGDS, but is still all-girls.
Now, all operating costs are paid by the Grand Duchy of Hallereton, so tuition is free, but it is very selective. Its students are mostly girls of particular academic or athletic talent who can't afford other top private schools or have other issues. There are, for example, quite a few student-athletes who were expelled from public school for behavioral issues but who thrive under the strict discipline of HGDS. There are also budding geniuses who have access to technology their public school couldn't afford.