The area was still known as "Kenya," formally "Protectorate of Kenya." Here are old stamps for the area. You'll note the "Kenya, Uganda, Taganyika" names on the stamps. They date from 1912 to 1935.
Wow! $2,800 for the 1922 Kenya KUT. Sure hope my family saved all those stamps from my letters in the old days from far flung corners of the world.