The "commute" from where you stay will take some extra time for travel (if you are going to the tourist areas you mention each day). But that might not be a deal breaker.
We went last summer and stayed near Kensington Palace in Bayswater. It was well worth it for what we wanted. We walked a lot of places or took relatively short tube rides. If we needed to we took Uber or taxis.
It was nice to feel like we were more part of the city. But for $600 you can do a lot to make up for that.