How To Beat "Good Doctor" with Eaze

Passive: You literally don't need to spend any money on gear or on powers. I will recommend sleep darts if possible.

Non-Passive: Just a single bullet will do, that's it, literally just a bullet or a bolt.

Step 1: Get to the Vault
In this one, we're not gonna focus too much on getting runes or clearing areas out. So this vault in question is right after the guy on the throne near the wall of light. You can try to make your way up through the window above or take the whale oil out on the right of that window. I wouldn't recommend the whale oil as if you're doing a stealth route, you'll lose points on that with how many people are there. Once in there, go on top the bookshelves. Watch for the walk pattern of the only guard in there. If you want, you can try to bend time to get rid of them and hide them on the bookshelves. However, with some clever teleportation, you don't really have to deal with them. Proceed to go downstairs, but DO NOT GO ALL THE WAY DOWN! There are two guards playing cards there, so instead, teleport onto the chandelier behind them. You have to be careful, sometimes they'll notice you.
Once there, you will be behind the elevator, which means the room will be on the left (right if you take the wall of light route). There are two guards and dogs in a separate room. There should be a room to your right when entering, go in there. Crouch when you open the door, if you haven't already, because there is a sleeping guard. Leave him asleep. There is a door at the end of the room, to which you can attempt to move past the guards to the back of the room. If you managed to do so, you'll see two rooms. Go into the room on the left-hand side. You should now see the vault I was talking about. You could do something I call "The Dishonored Lock Method" where you go through the last digit and once you've hit back on zero, you change the digit next to it and so on. We don't want to do that, as it wastes time compared to this method.

Step 2: The Save and the Code
What you want to do when you're at the vault is make a save file. It could be old or new, just make sure that you NEVER CHANGE IT PAST THIS POINT! You can now attempt to find Dr Hypatia. If you want a cheat right now, she's on the third floor on the right getting off the elevator. Once there, you'll see Dr. Hypatia doing work. Do what you wish, but what we're focused on is a patient in the back rooms. This is our key to unlocking the vault. He will tell you the code right before he dies.
Remember this code in whatever way you think best. Then right as he tells you the code, load up that save file I told you about earlier. Now that you have the code, just put it in. The reason this works is that the level is pre generated. Meaning whatever codes go to a vault was already chosen by the game before you loaded the map. Now that you have the syringe that should be in there (and a few other goodies), you'll now want go back to Dr. Hypatia. Do not go downstairs, instead go on the ledge right above her head. Then jump down and use your knock-out button. This will stab her with the syringe, making it probably the quickest passives of all the main bosses in the game.

Step 3: Finish
Do whatever else you want to do with this map (or if you're doing a stealth route for the first time, just don't bother). There are some hard placed runes, but they can be gathered if you know what you're doing. I didn't add this as a thing you need, but the power "Domino" works wonders for a passive run on this map in specific. Just make sure not to stab anyone with it. By the end of the mission, you'll not only probably be undetected, but also kill free for any of you wanting to gain those two achivements. As a bonus, you keep Dr. Hypatia, which gives you rewards in later missions.

Don't overthink this one at all. If you really need to kill The Crown Killer, just shoot at Dr. Hypatia and then perform a stealth kill. It works most of the time, but make sure you save just before you do it or else it just ends badly.