Try the following:
Basically, we want to add a password to the account you're trying to run the launcher with:
Open the Start Menu
Open Control Panel
Open User Accounts
Select the account that you wish to run League of Legends with
Verify that your account is a "Computer Administrator"
Select "Create a password" and follow the instructions in Windows
After adding the password, try running the Launcher program again. After updating, you may be able remove the password from the account and run the launcher without any problems.