Brook takes over from Gavin & Stacey actress Joanna Page, who is quitting the play in October.
Kris Marshall, Peep Show's Robert Webb and Ella Smith also star in the show, which centres around an overweight librarian.
"Kelly is absolutely thrilled," a source told The Mirror. "It's every actress's dream to appear in a lead role in the West End.
"She is passionate about the production and determined to nail the part. It's a huge break."