In June 2014, Taras Nykolyn, 46, pounced on Roger Maxwell as he took an early morning stroll near the Windmill landmark on Wimbledon Common in south-west London.
He forced the victim to the ground, smashing his face and breaking his wrist.
Then, while he was on remand at Wandsworth prison, Nykolyn killed Wadid Barsoum by hitting him with a TV, punching and stabbing him in their cell.
Ukrainian Nykolyn, of no fixed abode, pleaded guilty to both attacks at the Old Bailey with the help of an interpreter.
He admitted inflicting grievous bodily harm to Mr Maxwell on June 19 2014 and the manslaughter of Mr Barsoum on May 4 last year.
Alternative charges of grievous bodily harm with intent and murder were ordered to lie on file by the Recorder of London, Nicholas Hilliard QC, after hearing the defendant was suffering from mental illness at the time.
Prosecutor Simon Denison QC said: “Two psychiatric reports concluded that the defendant suffers from an abnormality of mental function, namely paranoid psychosis.
“They are satisfied that at the time of the killing of Mr Barsoum his responsibility was diminished.”
Although there was a possible defence of insanity to the attack on Mr Maxwell, the Crown was satisfied it was dealt with appropriately with the plea to a lesser charge.
Diana Ellis QC, defending, told the court that Nykolyn had been moved to HMP Belmarsh since the killing.
Then in November last year, he was transferred to Broadmoor secure hospital for an assessment before being sent back to the top security jail.
As the requirements have not been met for a hospital order, the defendant faces a jail sentence, the court heard.
Sentencing was adjourned to Friday, January 22.
HMP Wandsworth was built in 1851 and is now the largest prison in the UK, holding 1,877 inmates
Alongside HMP Liverpool, which is of similar size, the category B jail is one of the largest prisons in Western Europe.
The spot where Mr Maxwell was attacked is not far from where young mother Rachel Nickell was stabbed to death by schizophrenic Robert Napper on July 15 1992
In 2008, Napper pleaded guilty at the Old Bailey to manslaughter on the grounds of diminished responsibility bringing to an end the inquiry into one of the most notorious killings in modern British criminal history.