Update: I try to log in directly through the game (.exe file), and still it instantly disconnects me with error code: BLZ51900001. I tried logging into the game with my other bnet account and still its working. When I search the error code over the forum, I found a post similar to my situation: Can't log in (BLZ51900001)
According to the blue replying in that post, there was an issue affecting the authentication server. Could this be the case for my account?