Thanks you so much for translating! I've missed this story these past few months.
After this chapter I am certain that Koyou is a reincarnator and we are seeing her story from an outside perspective. This is completely modern medicine, and Koyou has clearly learnt from textbooks everything to do. Especially when it comes to the fact she was a student, and has done NO experimentation on her own to figure this out. She also has a modern understanding of sanitation, with cauterization and alcohol sterilization that a society without even surgery couldn't possibly know about.
Also very interesting to see the advisor start to see her dangerous personality. He doesn't know how to express it, but Koyou clearly can't emphasize with her patients and is only acting friendly superficially. We've already been getting warning signs, I'm looking forward to the day she does something completely beyond the pale and the Crown Prince finally realizes that flaw.
It falls under my suspension of disbelief, but doctors in those times absolutely wouldn't have syringes. Hypodermic needles weren't even capable of being made until the mid 1800s, metallurgy just wasn't there. Apparently in the 1600s they experimented with using feather quills as the needle and animal bladders as the syringe, but they didn't understand where to inject and sometimes killed the animals or humans they experimented on (it was also extremely unsanitary, of course). Surgery without a 19th century understanding of gangrene is also far more deadly, many of the soldiers in the Ameerican Civil War with treatable wounds were killed by clumsy surgery.