You can stay at the Travelodge at the top od Drury Lane for �105 per night. This is more expensive than most travel lodges and they do no reductions (mainly because of its location).
However, the 38 and 19 buses will take you from there to the BDC (stopping virtually outside at both ends). Buses run every few minutes. The views across the city from the top floors of the TL are great (ask to stay in the annexe across the roas in High Holborn - it has 13 floors). You can walk to Covent Garden piazza in five minutes. The "Oasis" swimming pool is virtually next door. It opens at 6:30am if you fancy an early swim. There are numerous good pubs, bars and restaurants within a few minutes walk. The rooms are clean and spacious.
I worked in Islington about half a mile from the BDC for about ten years. The area is OK - it's has become more gentrified in recent years. But it's a bit "Bohemian". I'd sooner stay in the middle of town.