I am not completly sure but if you open your logs in \League of Legends\Air\logs and find your most recent one you can see a lot of things about the games you played and one of them is your ELO.
It shows my ELO for ranked so I am guessing that the one for normal games is also correct.
1970 it says for me and you can also see the ELO of all the other players in the game, wins, leaves and how much ELO you gained/lost.
Edit: If it is hard for you to find it you can close your client, then open it again, play a normal game, then close your client again. Go find the last log and search your name in it.
Edit for people who like old threads: This doesnt work anymore since ages, thanks.