Hi! The statement or phrase "In the spirit of..." used in this situation maybe synonym to: "For the reason of"/ "for the sake of" ***but more (for the reason of) breach of promise, for ten thousand pounds!*** ***but more (for the sake of the) breach of promise, for ten thousand pounds!*** In the situation: [..."The people mentioned in the letter ask her to sell them the letter. She answers"] "If I sell the letter, I will do so NOT in the sense of blackmail, but more in the spirit of breach of promise, for ten thousand pounds!" The people mentioned in the incriminating letter was actually the ones asking her to sell the letter to them(and they are the ones who initiated), and maybe as much as she did not want to agree but she did. She asked for a high price maybe because for her it means she is putting her own integrity on the line. And that maybe her main concern, not the blackmail.

I googled it and as it turns out there's a thing a law as breach of promise which means breaking a promise of marrying a woman. The letter contains, among other things, a marriage proposal. Could that be the case?

I think she is thinking about blackmail, but doesn’t want to admit it even to herself. She’s going to give it a more legitimate name ‘breach of trust’, implying that those she would incriminate it deserve it, that’s it’s their fault (even if it really isn’t!).

A "breach of promise" means breaking someone's trust. "In the spirit of something" means that the person is motivated by that idea. So, although it's hard to understand this sentence without context, I would say that Mrs. Cheney is motived to sell the letter because someone broke her trust.