I went to the Docklands Museum yesterday, a fantastic place by Canary Wharf. It costs about £5 to get in but the tickets are valid for a year. I'll certainly be coming back. There's plenty of 'interactive' stuff for the kids and as much detail as any adult would like. It's in an old warehouse and has a fantastic feel about it. As I say, it's an undiscovered gem.
I've had very mixed reports from people who have taken kids - school parties generally enjoy it with an interpreter, but you need to know quite a lot of the background if taking your own/"borrowed" children around for them not to get a bit bored in some bits.
There is a photographic exhibition "Londoners at Work" from early Feb which looks quite interesting for history buff grown-ups.