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Hampton Court Palace (not a 'castle' but a terrific palace), Windsor. Then farther but manageable as a day trip: Leeds Castle, Hever, Dover, Highclere (would be the hardest to visit without going on a tour)
Hampton Court, Leeds Castle(though best from the outside, a stunning setting and exterior) or Hever (gorgeous setting, fascinating interior, Anne Boleyn's one time home and glorious gardens.)