Looking for a pet friendly flat

Hi!

I am looking for a flat/studio, whatever is available from about early January if possible.

Furniture doesn't matter too much, I've already got a few bits with me, I just want to make sure I can find a place where pets are allowed. I've only got a small ragdoll kitten, she's 5 months old right now and keeps to herself, she's discreet and rather friendly.

I am a 22yo man currently employed but making money through catsitting, writing, etc. I am also finishing up my personal trainer training which will eventually be what I'll end up doing.
Otherwise, I'm a friendly person, clean and tend to keep to myself but don't mind a good chat. I've got experience in home repairs/renovation from my past renting experiences if needed.

Zones 1-2 in London would be ideal but I'm open to slightly farther depending on the area. As for rent, I'm currently paying £740pcm but I'm open to up to about £850 as I really need to find another place. I can't stretch the budget too far though.

Please let me know if you have any questions, I'd be more than happy to answer them