When the client just disappears after you attempt to log in, it means your Windows clock is out-of-sync with the time on League's server. You can manually adjust your Windows clock by following these steps:
- Right-click on your computer's clock and choose "Adjust date/time"
- Select the Internet Time tab and click the "Change settings" button
- At this point you may receive a UAC prompt. After this prompt, click the "Update now" button to resync