In reality, Princesses Elizabeth and Margaret did dance at the Ritz but in a conga line.
The aria playing while Margaret suffers her second stroke is 'Sempre Libera' from Giuseppe Verdi's La Traviata. In it, the soprano sings about how she will resist her impulses to change her ways, which Margaret has just done.
Although Peter Townsend is shown in this episode accompanying young Margaret during her and Elizabeth's 1945 visit to the Ritz, he and Margaret had actually not met each other until two years later during a three-month tour by the royal family in South Africa, where his chief duty was to protect the younger princess. He had also been married to his first wife prior to all of this in 1941.
The story that Elizabeth reads to Margaret at her bedside is 'Sir Roderick Comes to Lunch' by PG Wodehouse.