The brutal crimes have shocked the largely peaceful island.
The 54-year-old man, identified as Lin, beheaded his sister with a kitchen knife outside their home in southern Kaohsiung city after an argument, police said.
Lin and his sister, 52, were both unmarried and lived together. He had been taking care of the mentally ill woman for many years.
"She was supposed to go to the hospital this morning and they quarreled when she refused," an officer said.
An eyewitness said the woman had knelt and pleaded for her life while neighbors screamed when they saw her severed head.
In the earlier killing of a girl, schizophrenic Wang Ching-yu decapitated the child in public near a metro station in Taipei last year, fighting off her mother and passers-by who tried to stop him.
He escaped the death penalty in court last week but was jailed for life.
He claimed that he had hallucinated he was a Chinese emperor from Sichuan province and believed that killing the girl would bring him concubines to carry on his family line.