If your League client does not exit when game ends for some reason, you can take this string, put in BAT file and run
> taskkill /IM "League of Legends.exe" /F
Find out if your name of game client is different from this default one (and place the right one).
Sometimes client crashed at the start of game, sometimes at the end.
After killing dead process you will be able to reconnect.
As I said, you can put it into Batch file (with BAT extension) and place some good icon on it, and name it like "reconnect to LOL".
The reason I recommend to place in into BAT file and not just "lookup in taskmgr" is that your screen could be blocked by league client. If it is blocked you won't be able to Alt-TAB. But you will be able to press WIN + execute and run the script to kill process (Win interface runs over any other process).
PS. I put it into gameplay section idk.