A friend of mine had issues with his league of legends launcher not starting, he only saw the logo and then it disappeared after a while.
He contacted the riot support who does whatever any support does, tells you to re-install the game and etc.
This is not wrong, it usually solves the problems.
But since you are reading this it is most likely not the case and you are getting more frustrated with their support.
The steps I would like you to try is.
1. Open the task manager and close all League of Legends applications running.
2. Run League of Legends as an administrator.
3. Download and run "ADWCLEANER". <---- This is most likely what you need to do
**_Close all internet tabs, all (possible) running programs, before launching the application. Then simply press "Clean and Repair" followed by "Restart now"_**
The current League of Legends client seems to be unresponsive when you have ad viruses installed on your computer (For some godly unknown reason)
**<<IF THIS DOES NOT WORK, FEEL FREE TO DROP YOUR DISCORD TAG AND I WILL TRY AND HELP YOU OUT>>**