I agree that League of Legends would be a great game to be played with only a mouse. You would just need a mouse with lots of buttons!
Take the Razer Naga for example (http://www.razerzone.com/gaming-mice/razer-naga). It has 12 configurable buttons on the side of the mouse, as well as 5 on top.
In League of Legends, you only require the following default key bindings on the keyboard:
(The following are crucial)
1. Q [champion ability]
2. W [champion ability]
3. E [champion ability]
4. R [champion ability]
5. D [summoner ability]
6. F [summoner ability]
(The following are less crucial, but almost required)
7. 1 [activate item 1]
8. 2 [activate item 2]
9. 4 [activate trinket]
10. B [teleport back to base]
Pressing numbers 3 and 5 through to 7 on the keyboard, by default, are to activate additional items, but it is very rare that you would have the need to activate more than 1 or 2 items in the game. I would say they are not required to be bound at all (or at least, I never press them).
Other useful keys to use in the game are Ctrl, Alt and Shift. They are modifier keys, which when pressed with the other key bindings or mouse clicks, allow the player to perform different tasks.
Of course, not all of this functionality needs to be bound to a key. Everything can be clicked with the mouse on the game's UI if need be. However, clicking the UI is rather slow, and I think it would be safe to say that a mouse similar to the Razer Naga would allow you to play League of Legends at a rather competitive level, with only using the mouse.