I'm not sure we know for sure however the debate as to wether Jack was right or left handed has raged for years.
Bury was right-handed. At Bury's trial Elizabeth Haynes provided the following testimony regarding an incident involving William and Ellen Bury:
“About eleven at night I heard her cry out in bed room. I went up and into their room. She was in bed. He was also. He was kneeling on top of her with a table knife in his right hand. She was continuing to cry out, and said he was going to kill her.”
“When a major serial killer case is finally solved and all the paperwork completed, police are sometimes amazed at how obvious the killer was and how they were unable to see what was right before their noses.” —Robert D. Keppel and William J. Birnes, The Psychology of Serial Killer Investigations