Kate, 42, was admitted to the private London Clinic on Tuesday for planned surgery, with her office Kensington Palace saying the procedure had been successful but that she was expected to remain for 10 to 14 days in hospital before returning home.
No details of her illness or surgery have been given, but a royal source has said the condition was non-cancerous.
Her husband Prince William, 41, who has postponed several public engagements while Kate recovers, was pictured leaving the hospital after a visit to see her on Thursday but did not speak to media.
Kensington Palace has said it would only provide updates on her progress when there was "significant new information to share". It added that Kate, who has three children with the heir to the throne, was unlikely to return to public duties until after Easter.
The announcement of Kate's surgery on Wednesday was followed quickly after by a statement saying King Charles, 75, had sought treatment for a benign enlarged prostate.
He is due to attend hospital next week for a "corrective procedure" and has also been forced to postpone a number of planned engagements on the advice of his doctor to allow for a short period of recuperation.
His wife Camilla told reporters on Thursday he was "fine" and "looking forward to getting back to work" as she visited Aberdeen Art Gallery in Scotland.
Usually the royals do not disclose details of illnesses, regarding all medical issues as a private matter, but Charles was keen to share details of his condition to encourage other men experiencing symptoms to have a medical check.
