Mulligan was detained for a minimum of 15 years after also being found guilty of Logan’s murder.
Logan's mum and stepdad were jailed for life
John Cole, 40, was told he would spend at least 29 years behind bars while Angharad Williamson, 31, would serve a minimum of 28 years.
The trio were convicted of killing the little boy in Sarn, Bridgend, South Wales, in April, following a trial at Cardiff Crown Court.
Passing sentence on Thursday, Mrs Justice Jefford said: “You are responsible for Logan’s death and all the anguish that has followed from it.
“Because he was killed in his own home, it is not possible to be sure what has happened to him.
“Shortly before his death, at which time he was three feet five inches and weighed only three stone one pound, he was subjected to a brutal attack.”
The judge described the injuries Logan had suffered and added: “Also the sort of injuries seen in abused children.
“Inflicting these injuries on a small, defenceless five-year-old is nothing short of horrific.”
Both Williamson and Mulligan were convicted of a further charge of perverting the course of justice – an offence Cole had admitted before trial.
Logan, a previously “smiling, cheerful little boy”, was discovered in the River Ogmore in Pandy Park on the morning of July 31 2021.
Police found him partially submerged, wearing dinosaur pyjama bottoms and a Spider-Man top just 250 metres from his home.
The youngster had suffered 56 external cuts and bruises, and “catastrophic” internal injuries, which were likened to a high-speed road accident.
Experts said the injuries could have only been caused by a “brutal and sustained assault” inflicted on Logan in the hours, or days, prior to his death. They also said the injuries were “consistent with child abuse”.
In the months and weeks leading up to his death, Logan had been “dehumanised” by his family, prosecutors said.
Caroline Rees QC, prosecuting, said: “He had been kept like a prisoner in his small bedroom in the flat you saw, a room likened by Williamson as a dungeon.”
Logan Mwangi
South Wales Police
Angharad Williamson, and her partner, John Cole, in the dock at Cardiff Crown Court
Elizabeth Cook