looking for the answer to an 19th century English riddle: "What king
had most cause to complain of his laundress?"

Dear underhill-ga,
This is a suggestion, but I cannot find substantial evidence to prove
it. It may have been Lady Eleanor Wall Oglethorpe who was laundress to
Charles II and later became involved in the Jacobite cause.

"Lady Eleanor Wall Oglethorpe (James' mother) was born in 1661 in
Ireland, but at age 17 became a maid to Madam Carwell in the court of
Charles II of England. In 1680, Eleanor -- or Ellen, as she was known
-- became head laundress to the king. In her new post, she was given
lodging at the rear of the palace -- opposite the quarters of a young
major in the Dragoons, Theophilus Oglethorpe. Before year's end, the
two were married. Their union produced a series of ten sons and
daughters beginning with Lewis in 1682 and ending with James Edward in
1696. After the death of Charles II and the Glorious Revolution of
1688, Theophilus and Ellen went to France to be with the deposed James
II. By 1696, however, they reconciled with England's new monarchs --
William and Mary"
http://www.hightowertrail.com/Oglethorpe.htm
http://www.cviog.uga.edu/Projects/gainfo/tdgh-jun/jun19.htm

The Oglethorpe ladies and the Jacobite conspiracies 
? By: Patricia Kneas Hill 
? Publisher: Atlanta : Cherokee Pub. Co., 1977. 
? ISBN: 0877970394 
Anne and Eleanor Oglethorpe, sisters of James Oglethorpe, who
established the colony of Georgia, were heavily involved in schemes to
gain the thrones of England and Scotland for Prince James, son of the
last Stuart monarch.
http://www.secondlifebooks.com/cgi-bin/secondlife/23277.html

As a matter of interest, where did you see this riddle quoted?
answerfinder-ga
Google Answers Researcher

I presume that there is some sort of wordplay involved, since that is
usually the case with a riddle.

How about King George III, who had to handle a "washing ton"?

King William IV would have complained if his laundress had tried to
remove "The Royal Tar."

ShrinKing.

I'm helping my mother do research for a scholarly book on Evelyn
Underhill, and this riddle came up in one of her letters to her mother
while she was at boarding school...unfortunately the answer to the
riddle is not to be found in the correspondence.

Thanks for the great info on Lady Oglethorpe.