The former Princess of Wales had sent the letters to late Cyril Dickman, who served as a head steward at Buckingham Palace for more than 50 years.
In them she wrote about how William "swamped" his new baby brother with "an endless supply of hugs and kisses" and how "naughty" Harry was constantly getting into trouble at school.
Cheffins, a Cambridge-based auction house, said the letters formed part of "a unique collection of Royal memorabilia".
The entire Dickman collection, which also included letters from Queen Elizabeth II and Prince Charles, contained around 40 lots which were estimated to fetch around 13,000 pounds.
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But they went under the hammer for around 55,000 pounds.
The Diana letters, split into six lots, were expected to fetch around 3,600 pounds, but went for a total of 14,900 pounds.
"The reaction to one tiny person's birth has totally overwhelmed us and I can hardly breathe for the mass of flowers that are arriving here!"
In another, dated 17 October, 1992, she says how both young princes "are well and enjoying boarding school a lot, although Harry is constantly in trouble!".
Some of the other items in the collection, sold by the Dickman family, included letters, cards and photographs from various members of the royal family as well as an unopened boxed wedding cake from the Queen's marriage to Prince Philip in 1947.
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