The sentence means the girl will be eligible to apply for release in five years' time, but will only be freed if she can prove to a parole board she is no longer a danger.
She has been described as 'selectively mute' and said nothing throughout previous hearings.
Sentencing her today, Mr Justice Soole said: 'The level of danger to the public is high. In the circumstances of your continuing silence, the critical question is whether there is any reliable estimate as to how long that danger will continue.
"All the experts are hampered by your silence."
The judge told the teenager that 'what precisely' happened on the field "is known only to you".
He said: "Further and most disturbing evidence points to this being planned and based on delusional thoughts."
The judge told the court: "The utter tragedy and devastation of all this needs no emphasis."
Katie was found with severe lacerations to her neck and chest on a playing field in York in January and did not respond to frantic attempts to revive her.
The court heard in July how the teenage killer was standing in a nearby cul-de-sac in York, covered in blood and carrying a blood-stained Stanley knife as she rang 999 to tell police what she had done.
Katie's mother, Alison, arrived at the scene shortly after the discovery and cried out in anguish: "She's killed my daughter".
She cradled her dying daughter in her arms.
The traumatised mother was led away from her daughter's body as paramedics arrived to take over the battle to save Katie's life.
The teenager who carried out the attack is understood to have had "irrational beliefs" about how people "may not be human and may be controlled by a higher and hostile force".
Prosecutors said the girl had talked of being convinced that people "weren't human and were robots".
She may have thought Katie wasa 'robot' when she killed her, a previous court hearing was told.
The girl had been under the care of the local Child and Adolescent Mental Health Services for a year before the crime, but, although her psychosis was being investigated, it had not been diagnosed.
When mental health funding was in the news earlier this year, Katie's sister Bethany tweeted: "All this talk about helping people with mental health. I hope they do! It's come too late for my family but hopefully it can help others."
Were warning signs over victim's mental health missed? Court told of worrying build-up to Katie's killing
The killer's mental health problems were described by one psychiatrist as the most complex she has come across in 20 years practice.
The court heard the girl had seemed reasonably content at home and was doing well at school until the start of 2016.
But she then lost most of her friendship group and developed an interest in the macabre and reported having suicidal thoughts.
The court heard these issues first surfaced in Christmas 2015 when the teenager began self-harming.
She was referred to York's adolescent mental health term at the end of 2015 or early 2016 and remained under their care until the killing.
During 2016, the teenager ran away from home several times and did so for a final time days before the killing.
Psychiatrist Dr Barry Chipchase, who reviewed the case for the defence, found evidence of "disturbed thinking" and 'thoughts that other people weren't human and they were robots'.
She received counselling by the mental health team, but not a full psychiatric assessment, even when prescribed drugs for "depression and anxiety" in December 2016.
A friend of the teenager told police she was 'nice but weird' and talked a lot about death.
The friend said the teenager 'had dreams about killing people' and she had 'bad thoughts' about Katie's family.
In posting a shocking self-harm Instagram photo 48 hours before the crime the girl was said to be "clearly crying out for help and support".
After the killing, police recovered a Lion King Simba doll from the teenager's bedroom with its ears cut off and a slash across the stomach.
