Kim's uncle Jang Song Thaek, 67, who had been called the second most powerful man in the country, was executed in December after being accused of attempting to overthrow the Communist regime.
South Korea's Yonhap News Agency, citing multiple sources Sunday, said since then there have been executions of all direct relatives of Jang, including children and the country's ambassadors to Cuba and Malaysia.
In his New Year's address, the North Korean leader said purges, including the execution of his uncle, helped remove "factionalists lurking in the party."
Jang's execution was seen as part of Kim's effort to consolidate his leadership in the isolated country since succeeding his father, Kim Jong Il, who died in 2011. Jang had been married to Kim Jong Il's sister.
"As our party detected and purged the anti-party, counterrevolutionary factionalists at an opportune time and with a correct decision, the party and revolutionary ranks were further consolidated and our single-hearted unity was solidified to the maximum," the younger Kim said in his address.
Yonhap, quoting one of its sources, said there had been "extensive executions" of Jang's relatives "including even children."
Sources told Yonhap the executed relatives included Jang's sister Jang Kye Sun, her husband and Ambassador to Cuba Jon Yong Jin, Jang's nephew and Ambassador to Malaysia Jang Yong Chol and the ambassador's two sons.
All of them were recalled to Pyongyang in early December, the sources said. They said the sons, daughters and even grandchildren of Jang's two brothers were executed.
The dates when the relatives died are not clear.
"Some relatives were shot to death by pistol in front of other people if they resisted while being dragged out of their apartment homes," a source told Yonhap.
The sources said some relatives by marriage, including the wife of the ambassador to Malaysia, had been spared but were sent to remote villages along with their families.