Hostel with ensuite bathroom in private room in good location in London?

Hello,

Will be visiting London for 12 days in October with my Mother, who is a senior (so not fourth floor walkups). Was looking to rent a flat on VRBO or AirBNB but also considering a hostel with an ensuite bathroom in a private room (did that in Sydney and was fantastic). We are looking for the following if possible:

1. Good location, easy access to the Tube and within Zone 1 where the main sights are located;
2. Clean;
3. Kitchen facilities; and
4. Own bathroom inside own room.

We try to stay in an apartment or hostel for longer stays to be able to prepare some meals. Most of the time we are out from morning to night so prefer to send our budget on sights rather than lodging.

Can you recommend and studio or one bedroom apartments on VRBO or AirBNB? And any good hostels?

Spent quite some time looking for places and am overwhelmed at where to stay...your help would be truly appreciated!