Officers discovered 16-year-old Cleveland Dodd in Unit 18 youth detention facility at Perth's Casuarina Prison in the early hours of October 12, after he contacted them through the intercom.
He was taken to a Perth hospital in a critical condition, where he later died.
Mr Cook on Monday said the situation was "not good enough".
"We've let that young detainee down and as a result of that we need to continue to learn and we need to improve," he said.
"What we do need to do is to make sure that we do better."
Mr Cook said Corrective Services did not currently have enough resources.
"We're hamstrung by infrastructure. We're hamstrung by availability of staff," he said.
"But we want to continue to work with the staff we've got and recruit others to make sure that we do better."
He said he didn't want youths to be detained in the maximum-security men's prison but it was necessary after a riot at Banksia Hill Juvenile Detention Centre caused extensive damage.
"Unit 18 is not what we want. It's not where we want but it is one of the necessities of our response to the situation in Banksia. Hill," Mr Cook said.
Cleveland was the first child to die in WA juvenile detention in the state's history.
His death is being investigated by the Department of Justice and the coroner.
The Corruption and Crime Commission has also commenced an investigation following allegations of serious misconduct.
Unit 18 opened in July 2022 to house the most challenging juvenile offenders after major riots at Banksia Hill.
Both Unit 18 and Banksia Hill have been plagued by high rates of self-harm incidents and detainees being placed into prolonged lockdowns.
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