For a step up from a Travelodge, try Premier Inn. Again they have family rooms typically a double and a sofa bed, and will supply a pack & play type cot, too.
The Carlson properties also have some larger rooms - they have some properties on the South bank, right next to the London Eye. Also some more further down the river towards Vauxhall - cheaper, but a bit less convenient.
Grange Hotels again have double/double configurations available, including a property right by St Pauls, but that's expensive on weekdays (often cheap at weekends).
Also the Hilton Tower Bridge has some larger doubles with a sofa bed & enough space for a cot.
Booking.com is also good in terms of finding the correct room configurations. It is obviously reliant on having been indexed correctly in the first place, but it's at least a start!