5n / 6d in London - 3n / 4d in Edinburgh - What to do??

Depends on your interests, but I'd say the following are a must for first-timers to London: Big Ben, Westminster Abbey, London Tower and Tower Bridge, St Paul's Cathedral, the area around Regent Street, the British Museum, Trafalgar Square and I'm fairly sure a few more very obvious one that I'm now overlooking. If you intend to spend 6 days in London it wouldn't hurt to get yourself a guidebook, though.
Brugge/Bruges wouldn't be the first city I'd think of when planning a day-trip from London, if only because it's about 3,5 hour one-way. It is a very beautiful city, but in your case it may be more worthwile to seek out a smaller historical town in the UK, for example Oxford or Cambridge, rather than spending a lot of money on the international train to Belgium.

What has living in the west village area of new york got to do with it?

Anyway, I'll second the OP, a day trip to Brugge from London seems like a ludicrous idea, have a look at train times on-line.

Paris would be a better choice if you want a day trip to another country.

As #2 says, a day-trip to somewhere like Cambridge or Norwich would probably be as rewarding from your POV as Bruges and much simpler/faster/cheaper (unless you have a special reason for wanting to visit Bruges, of course).

After your basic planning with the aid of a guidebook, you may find that there's no need or time for a daytrip out of London. You might consider a trip to Hampstead Heath or Greenwich or, going out of London, to St. Albans. And it's easy to get to Hampton Court, Canterbury and many other nearby places I'm not thinking of.

As for an area to stay in London, you'll have to rely on public transport, no matter where you stay. I like the Crescent Hotel on Cartwright Gardens, in the north of Bloomsbury. Others like the cheaper George, a couple of doors from there. Many people prefer the Kensington area. When you come up with a short list of hotels, check them out at tripadvisor.com.

I have little advice about Edinburgh, other than to stay as centrally located as you can. There are one or two distilleries in the city. I don't know about visiting them.

Regarding "surprises," make sure both of your passports are valid for three months after you expect to enter the UK.

-- Steve

How does one manage to have more nights than days? What date do you arrive on and what date do you depart on? From/to where?

If you are arriving in London and counting that as a day, you're kidding yourself. The same with the departure day. Those days are wasted.

If you have a total of 10 days including the arrival and departure day, that would mean you plan 4 full days in London, a day spent travelling to Edinburgh and 2 full days in Edinburgh followed by a departure day. Are you flying into London and out of Edinburgh?

You certainly don't seem to have enough time to consider trips outside of either city.

The south bank in london is really nice if its sunny- Tate Modern is free for general admission and worth a visit. Also go to Covent Garden is a nice part of town. Leichester sq near by isnt very nice but you can get cheap theatre tickets there if thats something you're interested in
I'd recommend stopping in York on the way to Edinburgh for an old city experience. Bath's also really nice but this is west of London rather than on route to Edinburgh- still its worth a day trip
Make sure you climb Arthur's seat in Edinburgh- its doesnt take very long and gives you fantastic views of the city and sea. If you're there in the next month there will be loads to do with the festival on!

The OP sent me a PM saying my response was rude. My, my. I thought I was pointing out a common mistake people make in counting days. But it seems the the OP has a way of getting 10 days out of only 8 nights. Miraculous.

I agree that Bruges is a whole trip on its own and massively out of the way. You don't say how you're getting from London to Edinburgh, but if taking the train, rather than a day trip from London, you might consider a stopover on the way north. York with its ghost walks and great rail museum right next to the station is a great town with lots of history or maybe Durham, a little quieter are stops about mid-way on the line. The train leaves from Kings Cross Station. If don't have a story on it, but you might look at the Sherlock Holmes Hotel on Baker Street.

Lots of London areas are a little like the village, each its own flavor. If coming from Heathrow and looking on the budget side, Bayswater near Paddington where the Heathrow Express terminates Kensington or Notting Hill. If more upscale Westminster - Picadilly is the heart of the city. The Thistle Victoria is a grand old lady brought back to life, a bit bustling area but vibrant, with busses to everywhere just outside the door at the station. Leicester Square to Covent Garden is the theater district for lots of restaurants and bars.