A short time in London, traveling with someone who is not enthused about cities, and wishing to use the Heathrow Express. Sounds like my first time in London. I have since come to love this city and recently spent three weeks there.
I am one of the people who contributed to the Little Venice thread that attracted you to that area. I agree with others that it is a nice place for walking---but not a great place to stay.
In the Bayswater area you can stay very close to Hyde Park/Kensington Garden, which together form a huge and mostly undeveloped park---just meadows and trees and paths, and a serpentine-shaped lake. You can access a number of great museums by crossing the park to the south side---the Science Museum, Natural History Museum,,and Victoria and Albert are all right there. Or head east through Hyde Park to Green Park, another green open space, connect to St. James Park which is just past Buckingham Palace. the streets you cross are busy but the park paths are traffic-free. St. James Park has beautiful plantings of flowers and trees, and more people than the other two. From the south end of St. James Park, you can easily reach Westminster Abbey and Big Ben, or turn east and head to Trafalgar Square in heart of London. This is how we move around London---using the parks as much as possible, along with the Thames path along the river. If it is too far walk we take the bus or the river bus; the tube is a last resort for me.
In addition to the hotels suggested by Rebecca above, you might look at Number 63, which is right across the road from an entrance into Hyde Park.
If you intend to take the Heathrow Express to Paddington, buy your tickets well in advance for the best price. Ninety days or more ahead is best of all. Depending on the hotel you have chosen and your amount of luggage, you may want to take a black cab from the station to your hotel.