I used to live here and go back very now and then definitely one of my fave areas. But the regular monthly rentals for sublets are at least £800 around here these days as it's tech-central (or Silicon Roundabout as we call it, after Old Street roundabout) with everyone working there pushing eastwards looking for better rents, you're more likely to get a better deal out towards (but not in) Stratford (trace along the canal looking for neighbourhoods). Whilst commuting will add to your costs it's still not much different than most coffees, and frankly anywhere along the canals are a delight to walk/cycle.
You may however find a summer sublet, take a ganders at GumTree, Craigslist and it's not entirely unheard of to see people on Couchsurfing list places (if there's a London flats group) I like to find sublets this way, but FB is indeed the obvious choice. (Potentially you might qualify in LondonStartupFlats…)
An alternative to these is The Collective, which is organised coliving (like WeLive) and just about within your budget yet includes a coworking space etc!
The pressure on cafés from laptop is enormous, if you rotate a few hours amongst a bunch you might get by, although power sockets are not common. There are endless awesome coffeeshops (such as The Department of Coffee and Social Affairs, Timberyard, Store Street Espresso, edgier little places like The Old Shoreditch Station … check out FourSquare of course, and there's also a bunch of WiFi/coffee apps), you really needn't ever set foot in a chain.
Google Campus is a proper coworking space and is free, if you get there first thing after they open you're pretty much guaranteed a space. Otherwise only if you're lucky…
London is not dodgy, you'll be fine anywhere north of the river. (And is one of the safest places to push your tolerance a bit )
If you're flexible on timings and willing to wait a bit longer, you'll find mid-month short lets but otherwise it'll almost entirely be month-to-month, however always ask the person leaving or going away might want some flexibility too. Late September onwards will be difficult (new semester) if you move your dates forward a bit it should easy getting rooms when students are away over summer. Personally I'd want to see the place, but some street view and a skype walkaround should pass muster.
A private sublet won't be a problem without a bank account just use TransferWise or whatnot to settle up, but make sure you'll get your deposit back similarly or as cash. Forget agencies.
Since the last time I was there it seemed everyone was using contact payments, I didn't once need cash. (Thus a bankcard with contactless is desirable, else a smartphone/ApplePay; works on transport too.)