This is a part of the optional quests Granny Rags gives you. Dr. Galvani’s flat is full of enemies, so getting through it without being detected and without killing anyone is somewhat challenging. If you take it slow and save frequently you should be fine, here is a guide to how I did it without killing anyone and without being detected:
There is a video at the end of this guide with a visual version of this guide.
The first room you get into contains two enemies:
Take out the one to the right first by choking him. Make sure the other guard is looking the other way. Leave the guard on the ground and quickly go around the glass cabinets to take out the other guard. It isn’t necessary to hide these guards.
There is another guard in the room to the right. Look through the keyhole and wait until he has his back turned, then rush in to take him out. At the end of this room is a strange room full of rats:
There are keys to the right of the door that open the door. When you do be sure to stay on the table, the rats will damage you if they can. Just wait a while and the rats will leave eventually. There is some loot in that room.
Be sure to explore this level carefully, there is a room to the left of the main room as well. On the next level you see a guard and a servant (?):
Just wait out their conversation. The guard will go upstairs and the servant to the room to the right of the stairs. Quickly follow her and you can take her out as she stands in front of the bed with her back to you. This level actually has a safe, I have already written a guide to opening it.
The last floor just has the guard walking around: