But, because I overthink things, it still took a while. There are loads of places to buy playhouses online - some are simple, cheap shells a bit like a shed; others are more luxurious than the average London home.
In the end I picked the Redwood Lodge Playhouse. I didn't want one that overtook our garden - and this one has a small footprint of one metre squared, but since its topsy turvy design means it expands as it rises, there's plenty of playing room inside.