Looking for a place for a couple and their kitty

My wife Asmaa and I are seeking a room or flat, ideally with an ensuite and bills included below 1400 per month. We're looking for a place to move in before end of March.

I work in Westminster while she works from home and is currently seeking other work in London. I'm looking for somewhere within a 40-minute cycle of Westminster or so, with a preference for being south of the river. Good train/tube links are a big plus, but the preference is to cycle on the commute.
We’re both 30 years old. We're both quiet and clean.

We are moving from a smaller town with the view to mutually climbing the career ladder, reconnecting with old friends, and hopefully making some new ones too. I like to occasionally host dinner parties and work on my charity, while my wife Asmaa enjoys mostly going to the gym and attending academic talks in central.

We have a well-trained, cute cat called Lavanta who has lived in house shares in Oxford and London. She likes a stroke and meows when she wants something, but otherwise is no trouble and is a Russian Blue with gray short hair (cat references from previous landlords available upon request).