a bit of a spoiler question

Both.

Tenei-in (the headscarf-wearing Shogun's widow) had to come up with a Plan B to get Ejima and Shingoro separated from her entourage, since she figured Ejima's maids were too loyal to leave them alone and spy on them. She asked that of Fujikawa and the handmaiden refused. Plus Tenei-in had every reason to believe that Shingoro would confess her whole plan since Miyaji told him everything. A fire would be just the distraction she needed. Crafty, no?

And Miyaji either a)hadn't been with a man since the late Shogun or b)was a virgin like Ejima. If you were a member of the Inner Court and you weren't the Shogun's mother, sister, daughter, aunt, or grandmother or a shaved-headed Buddhist nun-servant, you were chosen to be at his and only his, erm, service--which is why all the secret affairs with advisors and actors were so dangerous. Of course, a five-year-old master can be frustrating for the Inner Court. Miyaji was jealous of Tenei-in and her gigolo/actor, Gekko-in and Akifusa (a former actor), and Ejima and Shingoro--she figured she would be too old by the time the Shogun reached puberty. For a brief moment she was weak enough to ask Shingoro to have sex, then she thought better of it--she preferred keeping her head, as it were. So jealousy/shame played a part as well.