She said Dr Kelly blamed the Ministry for "effectively letting his name be known in the public domain".
"He said several times that he felt totally let down and betrayed," she told the inquiry.
Mrs Kelly learned her husband was the possible source behind the BBC's report on the Government's Iraq dossier as she watched television news with him on July 8th.
She said he had been withdrawn and queit that day and said: "He seemed a little bit reluctant to come and watch the news
"The main story was that a source had identified itself and then immediately David said to me: 'It's me."
"My reaction was total dismay, my heart sank and I was terribly worried because the fact that he said that to me, I knew that he was aware that his name would be in the public domain quite soon."
She said her husband seemed "desperately unhappy about it, really really unhappy, totally dismayed.
"He mentioned that he had had a reprimand at that stage from the MoD but that they had not been unsupportive, were his words."
Mrs Kelly said her husband had seemed certain that his name would become public.