I was torn there too. I do think 挟む is more for physical states (I’ve seen 合間 or 合間合間 used for “between times” ), so that was reasoning number 1 behind reading it as a room and not an interval of time. . The second being the pictures seeming to indicate passing through or by a room as she speaks, the 3rd being the timing. It just seemed weird to me that when she was going to bed, she would specify that it was between the times of cleaning soot, instead of it saying when she goes to sleep. This could be easily explained by her somewhat odd point of reference for life, being a shut-in living doll, though, so it’s the weakest reasoning. And my final reasoning was the particle used: You don’t usually use を with time words. I would expect に if 間 were to mean time (ala 間に合う).
I very well could be wrong, though, and I’m curious how a native would read it, so I’ll send this bit to my tutor and see what she thinks. When she gets back to me, I’ll post her thoughts here! 
Edit with her thoughts:
So, she seems to think it’s a room, and rather than the stairs necessarily being across from the soot removal room, it seems like the idea is more that the soot removal room is between two places. In this case, Kate’s room is the most likely candidate for one of those places, and the stairs are the other. So, you pass through the soot removal room from Kate’s room to get to the stairs, and then down the stairs is エミリコ’s room. (Though, from this description, the stairs would still be across from the soot room, if you are crossing through it to get to them, I would think, so not too far off base with that description).