That little house, designed to be a piece of Scotland, is left over from a big flower show at Exposition Park in October 1921.
Exactly where the house was in the park I'm not sure:
Oct 21, 1921,
Los Angeles Times @ Newspapers.com
It was repurposed as a tool house:
Jul 23, 1922,
Los Angeles Times @ Newspapers.com
Here's the tool house in 1924:
"Photograph of exterior of Tool House in Exposition Park with child standing in front, Los Angeles, 1924."
UC12031668 @ USC Digital Library