i think you'd like the area immediately adjacent to spitalfields market. i have a single, female, 31-year-old friend who lives there now in a modest 1 BR and she enjoys it. i think she's paying around £275 per week (plus you have to pay the "council tax" here when you rent).
the large majority of flats do come furnished here, incidentally, so that shouldn't be a problem for you.
being within a short walk to work would be nice if you're working long hours. and if you stick quite close to the market itself, the neighborhood is definitely a gentrified one. if you like indian/bangladeshi food you are also close to brick lane where a good, cheap meal is easily had.
again, i'd also consider islington if i were you. a tad bit further from your office than the spitalfields market area, but certainly not too far. a little quieter perhaps and still sort of eclectic.
another possible area would be shad thames. it's on the south side of the river, immediately to the east of tower bridge.
shoreditch is also an okay area, though i don't know it as well.
again, let me know if you have specific questions. i'm not sure if you are at the point where you are looking at particular neighborhoods or if you are merely trying to determine whether living in london would be a realistic possibility financially.