Holland Park's great.. peacocks, foresty bits... a nice cafe, some great play eqiupment.
Its near Kensington High street for shopping.
There's the river buses which are always fun, you can get off near the Tower of London and St Katherine's dock is a short walk away. You can pay to go in, or you can just walk around the outside. You can walk over tower bridge which is right there and see the gap where it lifts up when big boats come through. there's even a museum in the bridge if you want to pay.
Greenwich is good for a day out.. the cutty sark, a little market, the obervatory in the park where you can stand with one foot either side of the meridian line ... and the naval museum... you can walk along the river, and there are loads of nice cafe's
There are lots of free museums in london, and Time Out has a section on kids and family events, as well as free stuff.
Swimming's always good... might be a bit cold for the open air ponds on Hampstead heath, but the Lido in London Fields is heated, and it's near the museum of childhood in bethnal green, and the Geffrey museum ( both free entry I think)
Brick Lane Market at the weekend... with lunch at an indian cafe?
Columbia Road flower market on Sunday mornings
I'll be looking out for some more ideas on here later