A damaged past is a valuable commodity in today’s literary marketplace, dominated as it is by competitive self-exposure. The remarkable thing about Diski’s life-writing, by contrast, is its cerebral, anti-sensationalizing tone, the way it tacitly rejects the notion that early trauma retains an ineluctable hold on the self. Like Montaigne, she writes a curious brand of intellectual autobiography, in which her ideas and impressions — her response to reading “Moby-Dick” for the first time or her evolving views on psychoanalysis — are given as much salience as what, in more conventional hands, would constitute the main events.

In person, Diski is much like her prose: She talks about herself, and her present condition, with poised alienation. “It’s a unique experience,” she said of dying. “I’ve never done it before, and I won’t be doing it again.” Like the rest of her, Diski’s voice had taken a beating: froggy and phlegm-laced, a brittle truce between Received Pronunciation and her native London drawl, you can hear in it all the decades of cigarettes (Diski began smoking when she was 12) and, now, the lung cancer that has spread along her lymph nodes. Although the treatment seemed to have stalled the disease’s momentum, it had also worsened the pulmonary fibrosis (an incurable scarring of the lungs that, in effect, slowly suffocates the patient) with which Diski was diagnosed several years ago. It is now looking more likely that the fibrosis will kill her first.

Death, however, is only one-half of the story that Diski has been unfolding in the L.R.B. There is a portion of her overfull youth that, until now, has remained a blank. “If one cares about literary biography, it’s partly a duty,” she told me soberly. “At the same time, it’s a hunk of my life that’s not been there. And it’s irritated me, because there are inexplicable things. So I think it needed to be written about.”

In 1963, when Diski was 15, she received a letter from Doris Lessing, the future Nobel laureate, asking if she would like to come and live with her. At the time, Diski’s life was not going well. The previous year, after being expelled from boarding school — her offenses included climbing out her bedroom window in order to attend a late-night party in the nearby woods and stealing ether from the chemistry lab — she had tried to kill herself by swallowing a bottle of Nembutal. After her stomach was pumped, the doctors had her committed to Lady Chichester Hospital, a psychiatric facility. Of all the spooks and freaks who populate her past, the inmates she encountered “in the bin” were the ones she spoke of with the greatest fondness. All of them, she told me, “were narrators of one sort or another. They were all stories.” They were also, she said, extremely kind: “I felt quite contained and looked after by them.”

It was during her time at Lady Chichester that Lessing’s unlikely invitation arrived. “Ah, I see,” she remembers thinking. “That’s what’s going to happen to me next.” Lessing’s son, Peter, who was a classmate of Diski’s until her expulsion, had persuaded his mother to take her in. The head psychiatrist at Lady Chichester gave his blessing to the idea, and before long she was installed in the spare room of Lessing’s Camden home. Lessing got her new foster daughter into another school and paid for her to be psychoanalyzed. She also introduced Diski, already an aspiring writer, to the metropolitan literati. Having spent her childhood in a largely cultureless home, Diski found herself at the same dinner table as Ted Hughes, Robert Graves, Alan Sillitoe and R.D. Laing. “People sat around and drank wine and enjoyed themselves and had sex with each other,” Diski said, recalling the permissive, bohemian mood of the Lessing household.

Since Lessing’s death in November 2013, at age 94, Diski had been mulling how she might address the three years she spent as her foster daughter, and the ambivalent friendship they maintained for the next half-century. After the cancer diagnosis, it occurred to her that she might tackle the two subjects, Lessing and her illness, together. “Really it’s the form, that’s what excites me,” she said. “It’s a kind of pull-me, push-me thing.” Diski, who has written of her dislike for what she calls “straight autobiography,” has deployed the same double-barreled structure before. In “Skating to Antarctica,” she interleaves her Antarctic voyage with recollections of her early childhood. In “Stranger on a Train,” the atmosphere in Amtrak smoking cars (“a circumscribed space with a group of others all with our lives on hold”) keeps sparking memories of her time in psychiatric hospitals.

Diski describes this method in one of her recent L.R.B. essays. In the mid-1980s, Lessing came to visit her at Friern Hospital in North London, where Diski, who was then in her late 30s, was recovering from a nervous collapse. At the root of her breakdown, Diski says, was a sense of mortification at her failure to get any work done. Lessing suggested she simply write down her life story: “It’s interesting enough, and there are editors who can deal with sorting out your sentences and that kind of thing.” Diski, dismayed that Lessing thought her sentences needed fixing, recoiled from the simplicity of this approach. What she had not yet come to understand “was how writing gathers everything into itself to make a satisfactory piece. My story, someone else’s story, a place, an idea, a dream, human anatomy, the mind acting on the world, vice versa, some or all and more yet unthought of, had to be combined in the right amounts in order to make a book, an essay, fiction, nonfiction, history, comedy, whatever, work. I was enough of a writer to know that writing the story of my interesting childhood was not being a writer.”