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I was in London this week for 3 nights with 4 DC. You could stay in 2 rooms in a travelodge, it's what we tend to do, and would be nicer than the Dive where we ended up (which wasn't impossible, admittedly). And still walking distance to many sights (my idea of walking distance, anyway).
We would have stayed in Baden Powell House family room if available, could try there.