Cowboy with Keyboard makes good use of this idea. It's a type of shooter where you have to type to play, learning how to type without even realizing it.
It's set in the Wild West with arcade machines to shoot at, and the levels are increasingly difficult. You start with just a couple of keys and the space bar, and little by little, you end up using all your fingers to type with the entire keyboard.
There are western-style windows that align to represent the keyboard. Different villains appear inside each window, and you have to press the corresponding key in order to kill them. But you can't press all the keys at the same time, because innocent citizens will also appear inside the windows, and you don't want to kill them. If you miss a shot, there's a time penalty.
Besides three difficulty levels, there are also two game modes: standard and duel. In standard mode, you mainly use your reflexes, since the score depends on how fast you shoot. Because you don't have time to look at the keyboard, you end up learning faster. In duel mode, you have to type with different keys to write complete words in English.
Without a doubt, this is a great way to learn how to type. You can create different profiles, too, so different people can play at their own pace. You can also customize the score settings.
If you want to learn how to type and be a real sheriff in the Wild West, this is your chance.