UPDATE & FIX
A friend of mine browsed through some solutions on google and sent me one of this. Please keep in mind the first tip about closing any running applications in the background. For me it was steam wallpaper engine, which I run all the time, but for some reasons, after this new patch, I was unable to open league's client. After closing the application, I ran league normally and, thank god, it opened. For me it was a simple fix.
Here's the solution I read:
Close other running applications
There are cases when an application that is running on your computer can cause this problem. Certain applications such as Stream and Razer Synapse have been reported to affect the launch process of this game.
Right–Click on the taskbar and select “Task Manager”
Click on the “Startup” tab, click on any application to select it and then click on the “Disable” button.
Disable all the applications in the list and restart the computer.
Without starting any other application, run the “lol.launcher.exe”
Check if the League of Legends not opening issue still occurs.