I don't know if this is fixable but i'm very sad. I just can't connect to chat and i've tried everything I saw even changing the date didn't work Im starting to think i'm going to lose my account that i spent over $80 on PLEASE FIX THIS!
I've been having this issue today as well. I have no idea why it's happening and I've sent a ticket to Riot Support. Hopefully they can help but it is Riot Games.. 
PLZ I SPENT $80 DOLLARS AND I CAN'T EVEN CHOSE MY LANE ANYMORE! I'M SAD PLZ RIOT FIX THIS! I GOT A LOT OF CHARACTERS AND I CAN'T EVEN PLAY ANYMORE! I NEED MY STUFF BACK PLZ I WASTED TOO MUCH AND I DON'T THINK I CAN GET MY MONEY BACK
Yeah i don't know what to do anymore... I think they did this so they gain money and i have to make a smurf and spend even more money...
Ok... So i Did the time change thing and it connected BUT IT JUST SAYS LOGGING ON FOREVER*[zombie-nunu-tears]*
I just fixed the problem by leaving my client open while still connecting to the chat service. You should try this to see if it works.
