Zakaria Bulhan, 19, plunged a large kitchen knife into the back of 64-year-old retired special needs teacher Darlene Horton, and then "skipped" away with a "crazed smile" on his face, leaving carnage in his wake.
Sentencing at the Old Bailey, Mr Justice Spencer handed him a hospital order with a restriction order without limit of time.
The judge described the killing as "a tragic waste of life" and said the psychological effects on the survivors would never leave them.
He told Bulhan: "It is quite clear that when you committed these dreadful crimes you were not in your right mind.
Mr Justice Spencer handed Bulhan a hospital order with a restriction order without limit of time Russell Square attack Fri, August 5, 2016
A woman has died and five people have been injured after a 19-year-old knifeman went on a stabbing rampage in central London.
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US woman stabbed to death in Russell Square attack named as Darlene Horton
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“You were in the grip of mental illness, paranoid schizophrenia".
On August 3 last year, Mrs Horton was among many tourists on their way back to their hotels after enjoying an evening meal or shows in London's theatreland.
Initially, police feared that reports of indiscriminate violence at the scene of one of the 7/7 bombings could be a terror attack.
It was only later that it emerged that Bulhan, who is of Somali origin, was suffering from paranoid schizophrenia and was hearing voices.
Zakaria Bulhan 'skipped' away with a 'crazed smile' on his face after the attack
Within six minutes of the alarm being raised, armed police swooped on the scene and Bulhan was arrested without a shot being fired.
After being Tasered and bundled to the ground, Bulhan, from Tooting, south London, was heard to mumble repeatedly "Allah, Allah, Allah".
Bulhan admitted manslaughter by diminished responsibility and five charges of wounding with intent to cause grievous bodily harm.
He also pleaded not guilty to five counts of attempted murder, but admitted wounding with intent to Martin Hoenisch, Lillie Selletin, David Imber, Bernard Hepplewhite and Yovel Lewronski.
The pleas were accepted by the prosecution.
Bulhan admitted manslaughter by diminished responsibility and five charges of wounding with intent
he judge described the killing as 'a tragic waste of life'
Mother-of-two Mrs Horton was visiting London with her husband, Richard Wagner – a university professor from Florida who was teaching summer classes. She was due to return to their home in Tallahassee the following day.
She was described by a neighbour as "absolutely lovely" and "just one of the nicest, sweetest, family-focused people".
Mr Hoenisch, a retired fireman from Las Vegas, was visiting London with his wife, Laurie Kelly, when he was injured in the attack.
He said on Facebook afterwards that it was a "very unreal experience" and he was "still very saddened about the woman who was killed".
She said on Facebook she did not know whether to call it "luck or fate" that she escaped death, adding that she had watched as Mrs Horton "perished in front of my eyes".
Following the stabbings, a picture emerged of Bulhan as a "quiet, academic loner" who liked football, basketball and music.
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