The lady in question was Auguste Louise Gräfin zu Stolberg (1753-1835), and she seems to have been urged to the bold step of writing to Goethe by Metta von Oberg, a German baronness, with whom she lived in a cloister for noble ladies in a place called Uetersen, in Schleswig-Holstein, which was then part of Denmark.
Her brothers were Friedrich Leopold and Christian. They too were Werther enthusiasts, and, on their way to Switzerland, in May 1775, the young nobles stopped in Frankfurt and convinced Goethe to join them on their journey. Desiring to escape his entanglement with Lili, he did so. For the trip to Switzerland the brothers had clothes made to match the dress made famous by Werther: a blue frock coat, yellow leather knee breeches, and boots. Not only in their dress did the brothers cause a stir.
They were adherents of what was called the cult of nature and, near Darmstadt, took the opportunity to swim naked in a pond.
As Goethe wrote later of this episode in his autobiography: the sight of naked youths in the sunshine was no doubt a novelty but also caused a scandal. Well, we live in more enlightened times now. (Irony meter registering.)
Auguste fades out of Goethe's story in 1776, when he was becoming entrenched in Weimar and found in Frau von Stein a recipient for his outpourings. Auguste moved to Copenhagen in 1783 and married Andreas Peter von Bernstorff, an important government minister. The German-Danish writer Friederike Brun has many complimentary things to say about Bernstorff in her autobiography Wahrheit aus Morgenträumen. Brun's father, Balthasar Münter, was the Evangelical pastor at the court in Copenhagen, and it was he, along with Bernstorff, who accompanied Johann Friedrich Struensee on his ascent to the scaffold in 1772. Goethe, by the way, was one of the prominent European intellectuals of the time who wrote letters protesting Struensee's death sentence. An interesting portrait of this period in Denmark can be found in the novel The Royal Physician's Visit by Per Olov Enquist. Bruce Bawer, in his review of the novel, has a full account of what he calls "one of the strangest chapters in all of Scandinavian history."
The portrait of Auguste (now in the Goethe Nationalmuseum in Weimar) is by the contemporary Danish painter Jens Juelle, whose portrait of Klopstock is one of the most celebrated of that poet.