My first piece of advice is not to do a no-kill run as your first playthrough. If you just want the low chaos ending, a few kills are allowed.
If you are hard-headed: The doctor’s house has entries on multiple floors. The hardest way to go in “stealthy” is through the front door. Instead, try the balcony. You can approach the street over the rooftops, cross the street on the facade that stretches across access to the shoreline (crouch to go under the sign), and follow the rooftop vents to the balcony. There will be a single guard on the balcony, take him out with a sleep dart.
Once you go in, you will see 2 guards conversing on the other side of a big table, near a fireplace and door. They will finish the conversation and one will leave the room. You have a bit of time before the other comes back to choke the guard out and hide his sleeping body. Do it quick as possible before the other guard comes back into the room and you can take him out too. What you are looking for is either on that floor or the one above (the plague rat in a secret room behind a bookcase, IIRC, I’m going from memory here), I’d have to check a guide to be sure.
Or, if you are allowing yourself magic for this run, throw some points into possession, possess a rat, and go through the grate to the rooms that stretch lengthwise behind the big table. No guards will go in there. You can look through the keyhole to time the guards if you still want to take them out.
Or, you could skip Granny’s quest entirely, it’s optional. The game will give you non-lethal ways to resolve every story element, I believe Granny’s later appearance is left out if you refuse her request to poison the gang’s elixir.