Launcher won't update/Game won't play.

Since the 5.13 Patch update I've been having many issues with the league client. Firstly, when finishing champ select for a game, the game would "Start" and the client would disappear, the League of legends logo would not appear, and the game would not load the champion loading screen, I had to manually open the game window, close out of it, and restart the client before I would be shown a "Reconnect" button and finally be allowed into the game. As such I decided to try and repair my client, after doing such, the issue was still there and I figured an entire reinstall would be my best option, so today I deleted, and then proceeded to reinstall the client. I get to 99% at "Updating client" before I'm told that there is an unexpected error and to check error logs. Below is a list of the logs that I posted to google Docs. If I left out any logs that are of relevance or that need to be posted just alert me and I'll update the list. Hopefully someone can help with this issue.

EDIT: I had no idea that posting the Google Doc links would create miniature Google Doc Pages.

Logs :

https://docs.google.com/document/d/1So_dSOkClI0F2b-5VVSCB3_yhv_7iJKnbd3waDUJLJ4/edit?usp=sharing

https://docs.google.com/document/d/1OzYCDP6WnIBvQRY7c7vBzUeRDPmjaNPChZYHH7wbRiA/edit?usp=sharing

https://docs.google.com/document/d/12BYVPoKpGmx3yDHFeSJqmoyE65V0rocAJFmO5gpc2Wc/edit?usp=sharing

https://docs.google.com/document/d/1WsyYCdSyfnFrUFeuGccdBdoIETBl7foW7JOwG1zE0TE/edit?usp=sharing

https://docs.google.com/document/d/1OEJpLiwFR2caLeHQiLUYCsz1hMy0dQZ0UASSQSpjhMQ/edit?usp=sharing