Matt Bird writes all sorts of nonsense, but he dedicates a large chunk of his time to writing game walkthroughs.
Despite all the false starts and misleading solutions to your problem, you've finally gathered all of the Letters of Dee and, hopefully, have a recipe for the Panacea. Take the letters to the doctor in Buckingham Palace; he'll be overjoyed, believing he can make the Panacea. Leave him for now and go back to the Safe House.
Once you get back, Dr. Knight will radio in with some slightly bad news. Prepper will find out, freak out, and express his profound disappointment in you. After you're done listening to him yap, head to your container, and grab lots of guns, lots of bullets, lots of Flares and Molotovs, lots of everything. You're about to reach the point of no return, and you'll need lots of gear to get to the end of the game in one piece. Don't worry too much about empty space - you'll slowly run out of gear as you go along. Gulp. Upgrade your weapons to peak efficiency, then head for the manhole.
Go to Buckingham Palace. Dr. Knight's lab is now empty, and he wants you to meet him in the Queen's Quarters. Before you do anything else, check the codes you (should have) scanned on the various art pieces in the palace (one's in the art gallery, one's in the adjacent darkened room, one's in the main hall, the last is near the passage to Brick Lane Flats). Arrange the numbers into a code and use them to open the locked door near the Brick Lanes Flats path for a Capacity Upgrade.
Head to the art gallery again and check the retinal door. The doctor isn't here, so you'll have to find him. Head to the elevator and proceed to the second floor. You'll get a warning that this is the end of the game; heed it if you have anything else you want to do. Otherwise, ascend.
At the top, another of those danged electrical zombies will screw up your radar. Leave the elevator and head to the ballroom. You'll hear a lot of strange noises, and you'll find two zombies down here. Kill them off. You now need to get to the rear of the area again. You COULD do this the long way...
... or, from the ballroom, you could use the quick route via the spinning tube in the right wall, beside the stairs. Much easier, far fewer zombies to kill. Go through this area until you reach the room where you found the first book for Dr. Knight. Kill the zombie in the hall, then head down the corridor towards the large, two-level room. There are two zombies in here, one exploding; you know what to do.
The electrical zombie on the second floor here which will hide the red zombie on the lower level. Kill it for a Firepower Upgrade, then kill the next zombie beyond the door. Creep up to the second floor and take on the electrical zombie, as well as a normal zombie. The electrical zombie is no harder than in the nursery, though you may have to pursue it from one level to the other. Kill it off, then head through the door at the end of the pillars (which used to be blocked by fire) to find the Palace Kitchens. Scan hack the door at the end and step inside. (Note that you MUST kill the electrical zombie to get through here, as your scanner won't work otherwise. If you can't scan, it's not actually dead.)
Drop down over the debris in the next hallway and you'll start hearing Dr. Knight's music. Great. Just inside this room are three zombies, and one of them is a red zombie - Dr. Knight himself. Peg him in the head a few times to kill him before his two SWAT buddies come through. So long as you stay out of the room, they shouldn't all mob you at once. When they're dead, you'll have to pluck out Dr. Knight's eye, as you need it to use the retinal scanner downstairs. Tasteful!
Before you leave, you may notice a security door in the back of the room. Stand behind the piano and scan towards the other side of the room. You'll see the code written on the pillars and back wall. Use it to get through the door, then use C4 on the heap of trash. Go through here to find a Medium Medipack, a Firing Speed Upgrade and a Molotov. In the previous room, which appears to be a closet, you'll also find a vent. Go through it to find the actual kitchens.
This part of the walkthrough emphasizes it a lot, but it's worth saying one more time: if you don't have to use ammo, don't use ammo. You're almost out of opportunities to get more, and there are only more zombies to kill from this point on. Use the Cricket Bat whenever feasible to end confrontations.