League Client Won't Open

I play League on a 13-inch Mac with the OS system High Sierra (Version 10.13.5). The computer is old and was someone else's, but it was wiped and now I have it. As soon as the game was downloaded, problems abounded. The shop would randomly time out, and every couple of matches the game would crash. I could reopen it fine but it got annoying. I thought these problems were related to bad WiFi, but I am at my Grandparents house for the summer, and just now the Client wouldn't open. I didn't recently update the computer or any other software, and the WiFi here works fine (streaming Netflix with no lag etc). Last night I was able to play, but now I cannot. I used both Spotlight and Finder to open it, but the Client wouldn't even time out, the application just wouldn't open. No error, not even the icon appearing in Dock. I tried restarting, but it didn't work. Can someone please help me with this?

PS: Previously whenever I tried to open the application by clicking on the icon in Dock, it didn't work. I had to use the application folder or Spotlight. Is this just a glitch with the files?