Then I made the roof frame out of more cardboard (both kinds) and wood.
It's a separate piece, for ease of painting. A view of the bottom shows the (frankly rather rudimentary) structure.
Next step is the thatch. I cut pieces of a spare hand towel into strips and glued them on with overlaps. I wasn't too precise about measurements here, pretty much just eyeballing everything.
I realize the inspiration picture had way more thatching strips than this, but I only have so much patience.
Of course, my guesstimated measurements weren't perfect, but that's ok because I still have the corner bits of thatch to do.
And then here's how the roof looks back on the house. Only missing the watered-down PVA glue on the thatch to give it some weight.
And here's the version with the glue brushed on. Now it just needs paint.
Final note: if you're going to do something like this, consider investing the time/money to get a nice luxurious towel. The fiber loops on this one were pretty threadbare, so the thatch isn't nearly as textured or convincing as it might be.