I like how they bring up how they can't really force Olivia to do anything, because at the end of the day, she's practically an uber-powerful mercenary. Sure, she's got a title now but that's just a name. She has zero ties to the Farnest and can leave any time she wants. What are they gonna do? Put her in jail for desertion? Good luck; she'd kill as many as possible if you try. She's already a fan of murder.
But like any mercenary, the best way to keep her around and have her do things is to provide her with what she wants. She doesn't want money so paying her wouldn't do much, nor does she care about nobility so forget titles and land ownership. But history? Knowledge? Clues about her daddy? That's useful stuff. They can't provide that (they don't have it) but they can provide her with the means to get that knowledge e.g. the library.