I wouldn't have any problem with it - definitely budget but perfectly doable.
£35-£45 for a hostel is way overpriced; £10-£20 is a more normal range (or slightly more for a YHA).
Booked in advance, £40 a night would get you a (possibly non-central) Easyhotel, Travelodge or Ibis hotel. (Try laterooms.com to get a quick idea).
June might be better, as more University Rooms become available (see Universityrooms.com, currently showing ~£40/night for New Cross Gate in May; add £33 for the weekly travelcard, or a bit more if staying further out)
I've also never paid £50 to see a show - including at the National Theatre and Royal Opera House. Try tkts.co.uk or Lastminute.com. If you are flexible, I would expect to get something for ~£25.
Even in central London, you can easily find an evening meal for £10, or breakfast for £3-£5 (e.g; Wetherspoons, Wright's bar, etc.)