Why couldn't she just leave the room once she had delivered the message to the boss?
Because despite always saying she wants to stay out of the spotlight, she's immensely curious about everything, almost to the point like it's a disease. The gag is that same trait keeps getting her into trouble, and incompatible with her living a quiet life. This isn't out of the blue, it's why she became a scientist in her previous life.
In the novel, I also got the impression she felt like the jig was up so there was no point running away. Neil immediately told Vector who she was.
If she had that much IQ she would've made better disguise for herself long ago instead of keeping this same recognizable appearance.
Heck Vector himself is an example here that even just changing hair/eye colors are possible.
She has the disguise skill too. It's also more advanced than Vector's, so she's simply completely unrecognizable.
But she doesn't use the skill when she's at the blacksmith's out of convenience, which is her real mistake. As for just a general disguise...don't forget she's spending almost all her days at the blacksmith, and 90% of her time there is when she's in the forge, in the kitchen, or sleeping. Why would she put up with the inconvenience of a physical disguise when she has something far better and convenient for every time she goes out? She just got too comfortable being at the place and went to check on the master without thinking.
Also everyone jokes about her chest, but there's a serious component to that...people do really recognize her by that, and for some reason, the skill fails to cover that up. Implied to be because she secretly likes the attention. To her chest I mean.
Chapter was all fine, but I hope she goes her own path, or tells them to heck off or something, dont make it easy at least, its fun to see an antagonist character. If they make a "deal" based on her conscience or "otherwise you'd be a bad guy" - be the bad guy.
She really has no obligation to the master to take the request anyway, and he too was going to turn it down, so it's not even really a bad guy call in the end. She could still easily get out of this. But well, things aren't that simple later on...