For some reason, I was thinking it was $1000 for a week, which is relatively easy. $1000 for 10 days is definitely more challenging - it wouldn't leave much of a budget for tourist activities, although there are plenty of good, free, things to do.
You can currently book the EasyHotel in Croydon for <£23 /night for 10 nights in May.
For less than £40 / night you can get the EasyHotel Old Street, or Travelodges in Croydon, London City Airport, Ilford, Kingston, Wembley, etc. or Ibis budgets in Barking or London City Airport. To get good prices, these generally need booking non-refundable rooms in advance.
Presumably, at that price, AirBnB are only offering shared facilities - but I know that's not so important to everybody. Having access to cooking facilities would definitely help with the budget.
Exact location can be important. Not only are there some places that you probably wouldn't want to stay, but transport may be annoyingly slow.
I wouldn't normally recommend Croydon, but it has some advantages for a really tight budget. There are plenty of shops and cheap food close to hand, and pretty quick 24-hour access to central London.
Travelcard / contactless prices vary from £35 - £65 a week, depending on which zones you need, and are valid on pretty much all normal public transport options. (Bus, underground, tram, train).