"Statistics" is a fairly broad field. You could pick up a standard undergraduate probability/statistics textbook, flip to the later chapters, and have a source of new problems for months. (Later chapters on mathematical statistics are often more about proving and less about rote calculations, if you have the right book.) You haven't mentioned much of your background, so this might be appropriate and/or fun.
But since you specifically asked to avoid textbooks, let me offer an alternative suggestion: Lower-level actuarial exams. These exams are mostly about undergraduate probability, and there are a wealth of sample questions available online. Since these exams are taken under time constraints, your "kata" could be finding the fastest way to solve these problems.