The Duke of Exeter's daughter is not a high-ranking English aristocrat, but is a nickname for the rack, an instrument of torture, used in the 15th and 16th centuries and still in the Tower of London (here's a picture). The 2nd Duke of Exeter was an official at the Tower in the 15th century, hence the nickname.
I discovered this phrase earlier when I was looking at the entry for 'duke' in the OED. I also found another interesting phrase, 'to dine with Duke Humphrey'. This means to go without your dinner. It's origin is uncertain, according to the OED, but one suggestion is that, while some people were dining (in the early 17th century), others would walk past a tomb in St Paul's Cathedral, believed to be that of Humphrey, Duke of Gloucester, son of Henry IV. The tomb could not have been that of Humphrey, as he is buried in St Albans, but the name of an adjacent walkway in the cathedral was St Humphrey's Walk. The equivalent phrase in Edinburgh was 'to dine with St Giles and the Earl of Murray'' (St Giles is the name of Edinburgh's cathedral).