No matter how efficient the new chairperson may be, the meeting will not proceed (harmoniously, authoritatively) unless the new members cooperate

Ans - harmoniously

A party that cannot offer new ideas to deal with the pressing problems of the day must be considered politically (legitimate, bankrupt)

Ans- bankrupt

I am annoyed by the (haggling,patronizing) way in which they keep reminding me "how a well-bred person behaves"
Ans- patronizing

She may give the impression of being simple and uncomplicated, but we have found her to be unusually (sagacious, indisposed) in judging people.
Ans - sagacious

In the winter, Alice grows kale, as it is such a (hardy, sagacious) vegetable
Ans - hardy

Yay! All are correct!