...these were the virtuous prerequisites for vindication of some sort, for a triumph which would confound the sceptics...
...approaching some beneficent outcome which would make even my father's death assume acceptable proportions.
...I resigned myself to a lesson in reality which would be instructive but largely unwelcome.
The Bay of Angels, Ch. 1
The grammatical difference between that and which is subtle, and often inconsistently observed, even by the best writers. Kingsley Amis, one of the best, also a pedant, defined the distinction well in his grammar book The King's English (Harper Collins, 1997), adding that plenty of good writers have got it wrong from time to time while many bad ones have got it right. That Brookner's so Augustan prose isn't after all without its imperfections is one of the many adorable things about her. Jane Austen is another example of a flawed stylist - employing, for example, superlatives when comparing only two items, and probably mixing up her relative pronouns. (I'm unsure, not having read Austen for some time, and indeed in this context being somewhat loath to mention her.)