Setting aside how much time you'd want to spend inside some of the sights, I wonder whether you're adequately accounting for the time needed to walk between them. I checked Google Maps and found the following estimated times (based on 20 minutes per mile, which I consider reasonable).
Tower of London to Borough Market: 17 minutes. (Side comment: It takes some time to walk around Borough Market to see what's available, then to return to the stands where you want to buy your food. Plus you need to eat it, and you may not have a place to do so sitting down.)
Borough Market to London Bridge City Pier: 7 minutes.
Westminster Pier to Buckingham Palace (no exploring of the park or walking around Westminster): 20 minutes.
Buckingham Palace to National Gallery: 20 minutes.
National Gallery to St. Paul's: 32 minutes.
St. Paul's to south side of Millennium Bridge: 14 minutes.
Millennium Bridge to Tate Modern: 4 minutes.
Tate Modern to Aqua Shard Bar: 18 minutes.
That's 6.6 walking miles not counting getting to the Tower of London, walking around the Tower, or walking around the National Gallery (or potentially the Tate Modern). I walk a lot, and I spend a lot of time in museums, but I don't believe I've ever been on my feet that much in a single day. Then again, you may well be half my age!