I live in Central London.
It depends what you want to do. If you enjoy theatre, being able to stroll home is lovely, or being able to get up early to queue for day tickets for a sold out show at the National. And indeed knowing that you dont have to rush for a train because you can just jump in a taxi.
It also depends on how "edgy" you want.
My vote would be with Pimlico. Really central, good market, nice unflashy restaurants, not edgy, not too much traffic. A cental London village. You should have no problemgetting a two bed for your budget. (Note all lettings prices are negotiable and will vary by the quality of the tenant. A 45 year old couple couple wanting to stay two years, should be getting at least 10% off asking price. Try smaller agents like Dauntons who will provide a more personal service.)
But could understand why Borough might appeal. Quirky and buzzy, but a bit more mixed. Bits of Islington are nice, but the feel, to me, is affluent City banker/lawyer wife.
Or if you want a bit more on trend, you could look at where in Clerekenwell/Hoxton/OldStreet fits your budget
You will need a few trips to explore! Each area has its own feel.
If you are only here for a couple of years and can afford to live centrally, I would. There is no advantage in commmuting.
Central London is fun, and worth experiencing.