By far the quickest and easiest way finish this level is to do it the nonlethal way......(assuming you are on a low chaos run).....when you get inside the tower through the ducts, you will see the Lord Regent talking to the guards. Look up and to the right, and blink to that chandelier, from there blink to the upper balcony on the right.....check the door's keyhole for a guard than rush to his bedroom (first door on the left), get the audio log from the safe using 935, then SAVE YOUR GAME!!!! Blink on top of his bed, then wait till he enters the room to kill him.....cheevo popped!! Reload your save then just head straight to the intercom machine after you go downstairs to disable the pylons of course, then exit the way you came in......easy peezie!!!
IIRC: You can't do it the nonlethal way if you are on a high chaos run, my method is only to get this cheevo out of the way for peeps like me trying to get as much achievements as possible in one playthrough using the "save and reload method" while doing a no kills/no alerts run on their first playthrough.....
I just got this a few seconds ago..........I read about this achievement after I was already 75% into the Flooded District....luckily I had a save from when the boat was taking me to the tower, and I kid you not, it only took me 6 minutes to get from the boat to the Lord Regent to assassinate him......