She swore not to disclose a secret then she forgot and disclosed it by accident

Your wife is to be thanked for her honesty and her keenness not to disclose secrets. With regard to her disclosing the secret because she forgot, no kafaarah is required in this case, because in order for kafaarah to be obligatory, three conditions must be met: 

1 – The oath should have been sworn concerning a matter in the future.

2 – The person who swore the oath should have done so voluntarily and not have been forced to do so.

3 – The vow or oath should have been broken, by the person doing that which he swore not to do, by choice and remembering his oath. But if he broke his oath because he was forced to, or he forgot or out of ignorance, then no kafaarah is due. 

The evidence for that is the verse in which Allaah says (interpretation of the meaning): 

“Our Lord! Punish us not if we forget or fall into error”

[al-Baqarah 2:286]

And Allaah has said, “I have done that” [i.e., forgiven them]. 

And the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has forgiven my ummah for mistakes and forgetfulness and what they are forced to do.”. Narrated by Ibn Maajah in his Sunan, Kitaab al-Talaaq, 2043; classed as saheeh by al-Albaani in Saheeh Ibn Maajah, 1662, 1664. 

And Allaah knows best.