We walk and use the bus or boat in London (I hate riding the Tube). I have a bus map of London, but also like to consult this simple diagram of the main bus routes in Central London:
https://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf
You can see that in your area near Lambeth Bridge (which is just off the map) the 59 and 159 run north-south along Kensington Road.
For the Tower of London, you could take either to the Lambeth North Tube stop, or stay on the bus to Aldwych and transfer there to the No. 15 which runs east-west between Trafalgar Square and the Tower of London, also serving St. Paul’s and the Strand, with access to Covent Garden. This is a very nice sightseeing route, fun to ride upstairs in a double-decker bus.
Another way to the Tower of London, if you can walk 1/3 of a mile, would be to cross Lambeth Bridge and take the pedestrian Thames Path upriver (turning left from the bridge) a short distance to Millbank Pier. You can catch a Thames Clipper/Uber Boat there and have a lovely scenic boat ride to the Tower pier. With no traffic issues.
https://www.thamesclippers.com/plan-your-journey/route-map
The apps work great for a quick answer, but it is nice to have the big picture too, so I like maps. I bought this one at a London souvenir shop a few years back. Now that I see what it sells for on Amazon, I will take good care of it! (But there are several on eBay for half the price).
http://tinyurl.com/y9jamaua
And of course it could be out of date for some areas, so I always confirm with the TFL website.