Chief Inspector Donald Swanson in the writings found by relatives in 1980 identified Aaron Kosminski, a Polish man, as responsible for the brutal murders in London's grimy East End.
Kosminski, a Jewish barber, was among at least four suspects linked to the murders in the late 1880s in which women, often prostitutes, were found with their organs removed with apparent surgical precision.
The killer was dubbed "Jack the Ripper" after a man using that pseudonym claimed responsibility in letters to the media and police.
The Times of London Web site said Swanson wrote his comments in a book of memoirs of Robert Anderson, then assistant police commissioner.
Though other suspects have been identified, family members, who donated the book to the museum, said the detective felt certain he had cracked the case.
"My great-grandfather thought he got his man," Nevill Swanson said. "He would have thought he conducted his detecting job very well and reached a proper conclusion."
Mary Berkin, Swanson's granddaughter, said the case was commonly discussed by her family.
"It was general knowledge that my grandfather knew the name of the killer, and that there was no evidence except from a Jewish man who would not give evidence for ethical reasons," she said.
Kosminski also was named in a memorandum written in 1894 by Assistant Chief Constable Melville MacNaghten. He also named two other men as possible suspects.
Historian Keith Skinner, an expert in the Ripper killings, said there was no proof implicating any of the suspects.
"We don't know why these names come into the frame," he said. "Swanson's (notes) produce as many questions as they do answers."
Skinner said that what added to the mystery was a claim made in 1885 — and attributed to Swanson — that the killer was dead, while the man believed to be Kosminski died at a mental hospital in 1919.