In another investigation, the ombudsman found a cancerous tumour was incorrectly identified as benign (non-cancerous) - despite repeated scans suggesting otherwise.
The patient, a 54-year-old man, had his first scan in Tenerife, after becoming unwell on holiday.
His brother said: "When my brother collapsed in Tenerife, the hospital immediately identified the tumour for what it was [an aggressive brain tumour] and even offered to remove it.
"But my brother wanted to come home, he thought the best place for him to have the treatment was in the NHS."
The man returned to Gillingham, where the local hospital carried out further scans identifying the tumour, and referred him to specialists at King's College Hospital.
However, staff there "downgraded the diagnosis", saying the tumour was not cancerous. The man was not offered chemotherapy or radiotherapy.
He died in hospital, a month after an operation to remove the tumour, which was confirmed to be cancerous.
If the cancer had been identified correctly, he could have had surgery months earlier and potentially the option of chemotherapy and radiotherapy, PHSO said.
While the survival rate for this cancer is poor, the ombudsman's investigation suggests the man may have had a few extra months of life.
His brother said: "I wanted King's College Hospital to acknowledge their mistakes so that I can stop picking at the scab of trying to understand what happened to him and remember my brother as he was when he was alive."
In a statement, King's College Hospital NHS Foundation Trust said: "We fully recognise that errors were made in this patient's care, and we apologised to his family at the time of the incident.
"Learning from mistakes when they occur is crucial, and we have made a number of changes as a result of this case to improve the safety and effectiveness of the care our teams provide."
The trust has agreed to pay the family £3,500.