The former political adviser was in hospital for a year after being admitted with Covid in March 2020, with the illness causing serious damage to his body.
She said: “Boris Johnson actually wrote a really, really very personal note reflecting on things that he and Derek had been involved in together and his impression of him as a person, and also thinking of me and the children.
“And also Carrie as well, who I think has had a tough time, particularly when Boris was ill and she was pregnant with a first child.
“I think she has had a challenging time dealing with that. Mercifully, his health is great now but they must have had some dark moments as well during that period.
“So I thought it was lovely that he felt that he should and could and did actually drop me a note. So that was lovely.”
Kate also said her family are starting a “whole new chapter”, following Derek’s return home last month.
Derek is receiving 24-hour-care at home, which Kate has adapted for his needs.
“I believe he will improve. I think we’ve all been changed by what we’ve been through actually,” she said.
“The whole country has been changed by what we’ve been through.
“So I don’t think he’ll be exactly the same person again. I don’t think I will be actually or any of us will be.
“I don’t think you will be either. I think we’ve all been so affected by the pandemic generally.”