Which Neighborhood

I realize "where to stay" can become monotonous on the forums, so thank you kindly for reading and replying. I will be traveling with my 70 year old mother (very able bodied) in late September and need to base ourselves for 8 nights. We prefer a flat vs hotel to have a washer and a living space to gather and play cards, plan our day, just spend quiet time with each other. My mother cares for my father and sister who have a disability and this is her chance to focus on herself, not feel her normal worries back home. I've looked at a flat in Knightsbridge (Kinnerton Street), a flat near Victoria Station (a few doors away from Lime Tree Hotel...which is sadly booked). I've looked near Gloucester Station. We do not need a posh area. For my mom, I want an area we can walk out the door, feel a bit of the London spirit, restaurants and coffee shops nearby, shops and access to public transportation. I realize where to stay can be a tiring topic, but it is a large amount of money and I want my mom to simply experience the ideal spot to rest after a great day of sightseeing. Thank you for your helpful replies. I'm open to other streets/neighborhoods. Thank you kindly.