That's a tough one. With that many kickoffs, I doubt anyone else is trying to log in. I definitely would call LogMeIn. I would complain, I mean you're paying good...uh probably not, lol.
Have you tried Googling it and seeing what shows up? Also, have you run a virus scan lately. Like a 3rd party such as MBAM?
Actually just Googled "LogMeIn disconnecting" and received many answers.