There are so many fantastic museums, would there be one you haven't been too, which would fascinate both of you?
The weather forecast isn't looking too evil either, so you could possibly borris cycle it along to Greenwich, go to the market, the observatory is continually worth a visit.
Go for lunch someplace exceptional (there's always the Breakfast Club (Angel, Old St and other locations) if you're out of ideas) and to the Book Club (Shoreditch) for some table tennis? Go for a refreshing walk in Hampstead Heath (followed by dinner at the Chicken Shop/Pizza East round the corner)?
in no way underestimate the old-timey charm of feeding the ducks.
Brunch! Markets! Museums! Coffee Shops! Just wonder round Brick Lane/Shoreditch.
Picnics are good fun. Might hook a sunny day still. Holland Park has peacocks wandering around and a striking Japanese garden. Greenwich Park has the Maritime Museum along with Observatory too. They are showing the entries for the Astronomy Photographer of the Year for free starting on 19th I think. Depends what sort of stuff you two like!
DON'T be tempted by London Zoo. It costs a fecking fortune.
The lunch submission is good. Angel has a number of good places, or try the Riding House Cafe, it's just north of Oxford Street, nearer the Tottenham Court Road side.
Plenty of spaces do good deals for lunch. Check out toptable or similar.
Go to a one of the minor galleries: Saatchi, the Design Museum, William Morris gallery, etc.
Museums are always an option.
Kew Gardens is possibly my favorite thing.
