American Theresa Riggi, 49, who was serving 16 years in prison for the horrific crime, had only been moved recently from a facility in Scotland to the the institution in Nottinghamshire.
Riggi had stabbed Austin and Gianluca, both eight, and Cecilia, five - eight times at their second-floor Edinburgh flat.
She admitted culpable homicide on the grounds of diminished responsibility.
Riggi was held in Cornton Vale Prison, Stirling, before being transferred to secure psychiatric care.
It came more than two years after she was allegedly disfigured for life by a fellow inmate at Cornton Vale women's prison near Stirling.
Riggi was treated for her injuries in hospital after suffering a 40ft fall from the balcony of the townhouse in 2010, where twin boys Luca and Austin, eight, and daughter Cecilia, five, were found dead with suspected stab wounds.
It came several months after Riggi, 46, vanished with the children in July, after a break up with her husband Pasquale.
After she was jailed for 16 years in July 2011, Mr Riggi said: "The horrific manner in which my children died will leave an indelible mark on the rest of my life. As a father, my natural instincts were geared towards safeguarding my children from the dangers of this world.
"It pains me to the core that I was unable to protect them from the selfish, brutal and murderous act that ended their lives so unfairly.
"There is no justification for this heinous crime, repeated three times, nor is there any sentence that can provide justice for the overwhelming loss of three lives and the subsequent painful grief and devastation caused to surviving family and friends."
Jailing her at the High Court in Glasgow, Lord Bracadale said: "The result of these acts is a devastating family tragedy. The father of the children, Pasquale Riggi, and the wider family have been left utterly bereft by the loss of the children.
"And you, who had a genuine but abnormal and possessive love of your children, have lost them and are brought to this sorry pass."
He said that while Riggi's responsibility had been diminished, she was still responsible for her actions.
"The effect of the diminished responsibility is to reduce these crimes from what would have been exceptionally wicked crimes of murder to what are still very serious crimes of culpable homicide," he said.
Pasquale Riggi outside the High Court in Edinburgh Pasquale Riggi said there was no justification for such a heinous crime
"The number and nature of the stab wounds to each child is indicative of a truly disturbing degree of violence which, in order to bring about the deaths of three children, must have been sustained over a significant period of time. It is difficult to envisage the physical commission of such acts."
He added they were "ghastly and grotesque acts".