The following information primarily comes from the Royal Borough of Windsor and Maidenhead, also see the Durning Library:
Born the eldest daughter of John Benjamin Smith (merchant and politician) and Jemina Durning (from a wealthy Liverpool family). Note the spelling of the mother and daughter's shared first name; it's not “Jemima”. The family moved to London in 1848, spending the summers at their house in Ascot, King’s Ride. Her sister Edith married Edwin Lawrence, brother of James. Due to a childhood illness Jemina was never healthy and, after their parents’ deaths she stayed with Edith and Edwin, who kept King’s Ride using it for large social events, especially for the Unitarian community.
Jemina never married. In 1875 she gave money to the Great Ormond Street Homeopathic Hospital (later the Royal London Homoeopathic Hospital) and other medical establishments. Edwin Lawrence (who on becoming a baronet in 1898 changed his name to Durning-Lawrence) was appointed Library Commissioner for Lambeth which probably explains Jemina’s generosity to libraries. She (co-)founded both the Lambeth Durning Library and the Ascot Durning Library.
The Windsor & Maidenhead site says "Jemina Durning Smith died suddenly in 1901 and is buried in the family vault at Kensal Green." However, looking for her precise dates, we found this at the Newspaper Archive, London Daily Mail, 24 May 1901: "General regret is felt in Ascot at the untimely death of Miss Jemina Durning Smith, at King's Ride, Ascot, which occurred on Tuesday morning through leaping from her window. Deceased had been in ill health for some time past, and suffered from a spinal complaint. She died two hours after she was picked up. An inquest was held, and a verdict of suicide during temporary insanity was returned. "