Sushila Phillip died peacefully in her mother’s arms aged 36, her sister Holiday confirmed in a Facebook post.
Trevor, who was chairman of the commission for racial inequality, was also present for his daughter’s final hours.
Sushila had graduated with a first class honours degree from York University and worked as a freelance journalist.
Writing on Facebook Holiday said: “Dear all – my beloved sister Sushi passed away on Sunday morning.
“She went peacefully, in her own bed, in our mum's arms. Exactly how she would have wanted it.”
Holiday added that Sushila was “thanking everyone around her” in her final moments.
“Even until the end she was thanking everyone around her,” she said.
“After she was gone, I found a whole pile of yet unwritten thank you cards on her desk, we think they were to send to the many nurses who have cared for her. Whilst so much of her life was suffering, her legacy will be unconditional gratitude.”
In the interview the Duchess of Sussex discussed her mental health battles and said she had considered taking her own life.
"I have too much skin in the game to be neutral about the row sparked this week by the Duke and Duchess of Sussex", Mr Phillips wrote in The Times.
"And though I have thankfully never sunk to the despair described by the duchess, our family has spent more than two decades watching helplessly as my older daughter battled a severe eating disorder.
"Hours before writing these words she and I bade farewell on a familiar threshold: the specialist unit to which she admits herself periodically when the daily struggle against her demons proves just too exhausting.
"None of us can truly know what the duke and duchess experienced when they talk about the toll on their mental health."