The brutal crimes have shocked the largely peaceful island.
The 54-year-old man, identified by his family name 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 in southern Kaohsiung city. He had been taking care of the mentally-ill woman for many years, police added.
"She was supposed to go to the hospital this morning and they quarrelled when she refused. Lin then attacked her with a kitchen knife and severed her neck," an officer said.
Local television quoted an eyewitness as saying the woman had knelt and pleaded for her life while neighbours screamed when they saw her severed head on the pavement.
In the previous killing of the three-year-old 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.
Wang's case sparked fresh debate over the death penalty for child killers and there were protests calling for capital punishment.
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", local media reported.
Prosecutors called the crime extremely cold-blooded and demanded the death penalty.
But a district court sentenced Wang to life in prison due to his "mental handicap".