Jim Broadbent as King William

Sorry - but the repition of the word calls for clarification - its "dowager"

I think you're confusing people. :) Jim Broadbent played King William IV, who was married to Queen Adelaide - played excellently by Juliette Lewis (and you're right, she had a lot of common sense and was one of the most sympathetic people in the film!).
However, princess Charlotte, who indeed died in childbirth, was the daughter of the previous king, George IV and his wife Caroline.

~ History ~
In the late 18th century (the Georgian period), Britain was ruled by King George III, who would sadly later go mad. They made an excellent movie about this btw, The Madness of King George, with - iirc - Nigel Hawthorne as the King. King George III had several sons, the eldest of which was yet another George who would later become George IV. George III was very strict to his children and they thus rebelled a lot later on - pretty much all of them had illegitimate children; in fact, George IV was, for the longest time, the only one who had a legitimate child, the princess Charlotte. When George III was declared too mad to reign, his eldest son was named the Prince Regent (that's why this period is called the Regency period; Jane Austen may be the most famous person to live in this time - by the way, she HATED the Prince Regent). The Prince Regent really REALLY enjoyed being the de facto ruler of the country. He neglected his wife (who, you're quite right, wasn't exactly the most fit person install a sense of duty in her daughter - she loved to party and flirt and was reported to have had several affairs), even claiming that he only slept with her three times, and only paid any attention to his daughter if he could get anything from it. That is, if it made him look good in the press or if his daughter bowed to his whims.

Charlotte was the sole heir to the throne and much more loved than her father (well duh, anyone would look good against him). She was of course eventually married to provide more heirs. Her husband was the younger son of an impoverished German princely family - Leopold, later Uncle Leopold to Victoria and Albert! Charlotte soon got pregnant, but sadly she died in childbirth mostly due to the incompetence of the doctors (whose idea of medicine appeared to be mostly bloodletting). This means Britain was left without an heir to the throne, a King (George IV) who couldn't stand to be in the same room as his wife (and his wife was a bit too old to have children anyway) and a bunch of royal brothers who didn't have a legitimate child between them. Thus began the scramble for the brothers to marry wealthy, young heiresses, preferably royal princesses, to get children and produce the heir to the throne. William, later William IV, ditched his long-time mistress and married a princess Adelaide, who was about 20 years younger than he was. Adelaide got pregnant a few times but sadly she either suffered a miscarriage or the child died at a very young age.
Prince Edward, the Duke of Kent, married a sister of Leopold (Charlotte's widower): Victoire, a widowed German princess with two children (not shown in the movie). Victoire soon got pregnant and gave birth to a little girl: Victoria. Sadly, the Duke of Kent died when Victoria was only 8 months old, so she was raised by her mother, her governess and her mother's chief advisor, Sir John Conroy. Incidentally, Albert was the son of a brother of Leopold - Leopold was nothing if not cunning and savvy. :p

King George IV died in 1830 and his brother William IV took the throne. William was a playful, rambuctuous king who didn't like much pomp - his coronation was one of the cheapest in centuries and Adelaide actually paid for her own crown! William was a former sailor who had retained much of the no-nonsense habits of sailors. :) He spend the last few years of his reign just trying to stay alive so Victoria would be able to ascent the throne immediately and a regency would be avoided. Thankfully, he managed to stay alive up to a few weeks after Victoria's birthday in May 1837. :)

~ End of the history lesson (I know too much about this) ~

Also, his head was described as being in the shape of a pineapple - and Jim Broadbent even got that one down pat. XD

She was very wise and capable, I agree.

No, she wasn't from Denmark, and Charlotte was not her daughter. Charlotte was NOT "dotty" like her father.

The Queen Dowager was played by Harriet Walter.