To start with, why show the bit where Ciera ducks to avoid the assistant? To me, it doesn't seem to reveal much more in the way of character, and if it's foreshadowing for her ducking away from Lorn at the end of the chapter, I agree with Will777r that it would be stronger to get to that part.
You mention spending the last half-chime reading books cover-to-cover. Now, I know you probably meant that is Ciera's method of cataloging them, but to me it seemed like you were saying she could read all the new ones in that half-chime.
The scene with Denalia bugs me. First off, I thought that they already had a confrontation about the reports in Chapter 6? (I just checked, and they did cover some of this.) She didn't once ask Denalia if she'd checked the reports for accuracy -- which is one of the first defenses of people who believe they've done the work required and are being unfairly targeted. I'm willing to accept she blindly claimed new methods for doing them, since she felt the need for some explanation.
My main complaint, however, is that it seems to me like Ciera gave in to Denalia far, far too easily. So easily I saw it as a plot contrivance to keep Ciera there late. If it was a half-time until quitting, not even Denalia could expect Ciera to finish before leaving (and she doesn't), so it is clear Ciera is being asked to work late to do so. Does she have no power to negotiate back? At all? The board is mentioned, but surely if she was able to do the inventory during the next day (instead of overnight), the board would still side with her rather completely. Beyond that, there could be any number of reasons she was able to work faster (even with her "new methods") in that particular section that she could use as reasons to get more time. She's more familiar with that section, she was already building off of previous work she knew had been done, etc., etc. She doesn't show the least hint of her mouth talking back like a few minutes before, or anything, and I expected something there.
You're the author, so I imagine you can figure out something to do better than I can, but that was a spot where I really thought the characters were puppets with the strings being pulled for the plot, rather than naturally leading into it.
Another small note on chronology: You have the beginning of this chapter after we've "seen" Lorn get into the library, but it seems to go back before he even makes the attempt to get in. If there's not an overriding reason otherwise, I generally prefer to read in chronological order, and this messes it up. It took me a bit to reassure myself that this wasn't the next day, with Ciera (or someone) going to discover a bunch of books missing. This was despite the fact that I recognized the setup for having Ciera meet Lorn.
I also thought Ciera was too chatty and helpful with Lorn, but my problems started when she discovered a man in the library. A Man! In The Library!! But she has no immediate reaction other than "he's a little short for a man, with a lean build and elegant demeanor. Oh, let's look at his clothes and figure out what he does for a living." If men can't get into the library, that should be enough to give her a good jolt of surprise, if not shock, just right there. I don't think it was the best time to give the description, at least -- and anyway, we all know it's Lorn. Who else could it be?
I liked the cleverness of using other paper instead of the books, but like the others I also wonder about the sounds and the availability of paper.
Is there any particular reason Lorn would give her his name? Especially before they make it out of the library? And why his name, rather than a fake one?
Also, Lorn should have torn and bleeding hands in this chapter, after the end of the last one. Or bandaged ones, or something.