A possible fix for when the game does not load and you have "game in progress" in the client.
When the game does not load and the client just says "game in progress", do NOT close the client.
Open up task manager.
Find the process called 'League of Legends (TM) client' - end that task, and ONLY that task.
Now you should be able to click 'reconnect' on the client to join the game.
I hope this helps.