Ms Langdale asked Sir Duncan about an email he sent to Dr Jayaram on 25 May 2017.
The email said: “I want you and your consultant colleagues to know how deeply sorry I am for the personal distress that you have and are all suffering and for my part in not intervening sooner.”
On the same day, Sir Duncan went to find Dr Jayaram in person, the inquiry heard.
He told the inquiry that the two men had an emotional conversation, and added: “I remember us putting our arms around each other.”
Sir Duncan told the inquiry he now appreciated there was a “huge amount of sympathetic support” given to Letby by senior managers of which the board was “not sufficiently sighted of”.
The inquiry heard Letby’s parents, John and Susan Letby, sent a letter to Sir Duncan in July 2017 describing their daughter as being “singled out for punishment”.
The letter called for a meeting “as soon as possible” because “the anguish this situation is causing has become intolerable”.
Sir Duncan said he did not respond to the email, which came after the police investigation into Letby had begun.
The inquiry has previously heard evidence from the families of Letby’s victims that they were “kept in the dark” by the hospital.
Asked by Leanne Woods, representing some of the families, where they fitted in the “big picture”, Sir Duncan said: “The families were not in the big picture.
“We did not exercise appropriate duty of candour towards the families and that was a failure.
“A serious failure.”
The inquiry, now into its 12th week, is expected to sit into the new year.
Letby was convicted of murdering seven babies and eight counts of attempted murder, including two attempts on the same victim.
She is serving 15 whole-life prison sentences.
