need an advice 'bout London

hi there)

me, my hubby and our little daughter are moving to London next spring. i'm trying to  select a region to live,   and my mind is about to blow up) my hubby will work at Liverpool Street Station, so we need a 1 bed apartmet not far from the city center (2-3 zone), in non-criminal region with a park and malls. is it possible to find smth. on 1000-1200£  per month? what region can you recommend?

thank you!