[GAMEPLAY... sort of] Mouse in-game is buggy

I have a perfectly functioning logitech mouse. In every application including my desktop, google chrome, CS;GO, WoW, Hearthstone, and more, my mouse works completely normally. However, in League of Legends, it acts VERY peculiarly.

The first and most annoying issue is that it teleports across the screen randomly. I can move it to the right, and it moves normally. Move it to the left, moves normally. Move it back to the right, it teleports to the right INSTANTLY with the tiniest touch. This happens randomly and in all directions.

The second issue is that changing the "mouse speed" option doesn't change my mouse speed. I'm pretty sure this is the cause of the teleport problem but I don't know the cause of this problem. It goes the same speed on 0, same speed on 100, same speed on 50. Changing the speed does not seem to change the frequency or severity of mouse teleportation.

I basically can't play. I've cleaned the mouse, changed the batteries, taken it out, shut down computer, and none of it works.

DIRECTIONS -

1. Plug in mouse
2. Move mouse
3. Observe that mouse teleports
4. Change mouse settings
5. Observe that mouse settings don't change mouse speed
6. Rage

Obviously these directions have no use in the purpose of recreation to anyone but myself, but it says to do it in the "How to Report a Game Bug" thread and summarises my problems well so I thought I should.

When recording my computer kills itself so I can't show you a video, and screenshots wouldn't help.