Launcher Doesn't Work

So I updated my computer for patch 5.12, and did not immediately log in because I had errands to run. So later in the day, when I tried to log in, the launcher either did nothing after showing the League of Legends logo, or I got a message saying that I already had an instance of the launcher running and asking if I'd like to cancel that instance of it. The end result is that I can't get to the first window for League where you can find the log-in button, the patch notes, and other League-related news, and, consequently, I can't log into or play league.

I've tried restarting my computer, using the task manager to make sure that the launcher is definitely not working before I try again, and even uninstalling and then reinstalling League.

I'm not really sure what's going on or how to fix it. Thoughts?