The answers to my post were indeed very helpful, thank you all! I unfortunately had really only two partial days as I had to work my than I anticipated. I stayed at the Andaz London Liverpool Street, which was a very close walk to the Tower of London, Tower Bridge, St. Paul's Cathedral, and for my purposes, where I needed to be for work. I loved the hotel and would definitely stay there again. Free wifi, free non-alcoholic minibar, free wine and hors d'oeuvres every night, and the rooms and stay were great. Also right next to Liverpool Station so you can take the tube anywhere.
I landed just before 7 am on Saturday morning. Took the express train to Paddington Station, cab from there (could've easily taken the tube too). Being so early, my room wasn't ready, but the wonderful staff at the Andaz gave me access to the spa/health club on the premises, so I got a refreshing shower. I them immediately went to the Tower of London, which I enjoyed immensely. Walked across Tower Bridge, and had lunch at The George in Southwark, walked back over the London Bridge (and saw the makeshift memorial to the victims of the recent terrorist attack - very upsetting). Worked for several hours, caught Evensong at St. Paul's Cathedral, then worked more.
The next morning, I got an early start and walked from my hotel, mostly along the Thames (saw the Eye on the other side of the river), to the Churchill War Rooms & Museum (which opened at 9:30). This was an absolute highlight for me. I would rate this as one of the top museums I've ever been to. Get there early if you go. I had no wait when I entered, but after roughly 1.75 hours there, there were, by my estimate, over 200 people waiting to get in.
From there, I took a brisk walk around the very nearby Westminster Abbey, Parliament, and Big Ben. I wish I had time to get into Parliament, but sadly, I didn't. I walked to Buckingham Palace, through St. James' Park to the Mall, which I followed to Trafalgar Square, where I had an enjoyable lunch at the Admiralty. The rest of my trip was spent working, with the exception of drinks on Monday night at the rooftop bar at Sushi Samba, right across the street from the Andaz.
If you stay at the Andaz or in that area, I would highly recommend getting dinner at Dishoom in Shoreditch. Excellent Indian food, at what is, by London standards, reasonable prices. They have a beautiful veranda you can sit outside and enjoy. Just plan on a wait.