Looking back at the the OP's original thread, he says that while he's riding, his wife and children will be sightseeing. Where do *they* want to be for that week?
To me, if I'm visiting a city I want to be fairly central. If I'm sightseeing, I want to be able to walk easily to at least a couple of major tourist attractions, and be a *short* public transport ride from others. I want to be able to have dinner somewhere central and again, walk or have a short journey back afterwards. If I'm close enough or the city is small enough that I can walk all day and take in a lot of the sights, so much the better.
For that sort of trip I'd rather have a small room in an adequate, clean, central hotel than a suite in a palace significantly further away - I'm not going to do much more than sleep and have breakfast in the place.
I'd wonder about some of the small independent hotels in Bloomsbury, near Russell Square - family have used different ones in the past - or maybe look in Bermondsey (up and coming, near the South Bank) or even in the City - a friend has had good deals at hotels near St Paul's and the Tower (the Grange, I think).
Of course, that all depends on how old the OP's kids are and how keen they, and his wife, are on routemarches around the sights ... If the Aquatic Centre in the Olympic Park is important, then maybe we're back to Stratford in any case.
(I had a quick look at Premier Inn prices, just out of interest. A family room for a week, 28/07 to 04/08, is £1100 in Stratford(!!), £900 near Tower Bridge, and £650 in Greenwich.
You could probably do a lot better, and a lot more central, for £1100 ... But Greenwich might be a good option - big parks and some tourist stuff (Royal Observatory, Maritime Museum, Cutty Sark tea clipper, cheesy weekend market, Old Royal Naval College), quieter than central London, easy access by train or by river boat to the centre.
And while access to Loughton isn't ideal, it's straightforward by DLR and Central Line, and I'm close enough to offer an escort if the OP wants to ride up to registration or the start.)