If the client just doesn't open, start with the basics: check that your system meets the minimum requirements, then run the Riot Client as administrator. If that doesn't help, go to Task Manager and kill all Riot and Vanguard processes, then relaunch. Corrupted installation files are another common culprit — use the built-in repair tool in the client settings.
A spiking connection mid-match usually isn't the server's fault. First, switch to a wired connection if you're on Wi-Fi. Then check whether other apps are hogging bandwidth in the background — Windows Update loves to kick in at the worst times. In the game settings, manually set the server region closest to your physical location. If ping stays high, run a traceroute to the game's servers to find where packets are getting dropped.
If you can't get past the login screen, first verify your credentials at account.riotgames.com directly. A failed login loop sometimes clears itself after deleting the cache folder located at %localappdata%\Riot Games. Two-factor authentication issues are common after a device change — check your email for a verification prompt that might have gone to spam.
When the queue just spins forever, the issue is almost always a region mismatch or a corrupted session. Sign out completely, close the client, delete the lockfile in the game directory, and relaunch. If the problem persists across multiple restarts, the matchmaking service itself may be under maintenance — check the official status page at status.riotgames.com.