Second attempt to reply (first attempt somehow disappeared completely)
I've never done the hotel thing in London - either lived there or close enough, or stayed at mum-tat's place, and no she can't do B+B for 3. I get the impression some London hotels work on the basis that guests from around the world are unlikely to come back, so they don't try very hard - although the online review era may have weeded out the worst. AirBnB can be a lottery - not sure i'd recommend it with kid in tow. It may be safer to go for one of the chains rather than 'quirky'...
Somewhere touristy may be out of price range, and (assume we're talking a weekend) things may be expensive if one of the big london football teams has got a big match on.
danski 's comment about the City may also apply to places that cater more for business travellers in the docklands patch, although again a big event at excel or the O2 may bugger that idea up.
as a slightly random suggestion, would somewhere like woolwich work? it's got the Docklands Light Railway and Crossrail and a fairly frequent normal train service (all subject to disclaimers about engineering works), so is fairly well connected, and (if it's working that day) has the woolwich ferry and the pedestrian tunnel under the river, which may appeal to kid. and handy for blackheath / greenwich park (and the museums) which may appeal for the other day if they don't want to do central london again.
Woolwich doesn't have a stand alone clock tower, but would the
royal arsenal gate,
town hall, and
former co-op head office / department store (now a travelodge) do for a start? The only churches I know of are two former 1930s cinemas.