Does informing the husband of his father's health condition, after the wife swore to her sister-in-law not to inform him, entail an expiation for an oath (kaffarat yamin) upon her or upon the sister-in-law who made her swear? Is this considered a breach of promise? And what should she have done?

Expiation is incumbent upon the one who swore the oath, not upon the questioner, and it is an expiation for a broken oath (kaffarat yamin). If the oath was uttered unintentionally, it falls under the category of 'laghw al-yamin' (idle oath), for which no expiation is required. The questioner did well by informing her that she would not abide by the promise, and she did not break the promise since she never made it in the first place. Informing the husband was not obligatory for her, as fulfilling the promise made to the swearer is commendable but not obligatory.