"We want to stay at a good London hotel ($100-$160 a night)."
Each traveler's definition of "good" hotel is different--but in general $100 or thereabouts is low for a good hotel in London. If you mean clean and decent, that's one thing, but what you consider a good hotel may not be obtainable in London at the low end of the price range you gave.
Unfortunately, London is one of the more expensive hotel cities in Europe. Quick examples (obviously it is only an approximation to give costs in dollars due to changes in the exchange rate) Rick has published average London hotel cots, in 2009, as follows: average cost of a double room at a "mid-range tour hotel" in London averaged about $225, with a $330 average for a double rm at "upscale business-class" hotels, and $120 as an average for a double room at "budget hotels."