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.”