If you want to live alone, you can find 1 bedroom flats for ?1300. Studios you may find cheaper.
I am looking for rent myself so I checked zoopla extensively and I can say that the best and cheapest place to live would be Victoria/pimlico. They have cheap studios in that area and also it's very busy area because of Victoria station, it's near vauxhall, near Knightsbridge, near Mayfair or south Kensington. And it's still in zone 1. I heard that council tax is cheapest there?
Or you can rent 2 bed flat with other girl and then maybe you could only pay ?800-900pcm each
The most important thing is to find good place with intercom, but also if you are looking for studio, then make sure you have a bit more space and you have where to put/to hide your personal stuff/documents, etc.
But the other thing is I am not sure where you will find private landlords. In London it's very hard to find it. I could suggest you maybe joining Facebook groups and asking them? Should more private landlords there.