I've used logmein for around 8 months or so on my mac. Since upgrading to OSX Lion I didn't have any problems. But it seems that this problem has started a few days since updating to 10.7.1 last month. I can get into the GUI of logmein, but as soon as I press turn on the whole interface completely freezes. You cannot click on anything, but the app is responding (as in, it isn't actually frozen). The service doesn't start. I have tried completely uninstalling the software using the uninstaller, and also using appdelete - no joy. I have removed the computer from the account on the web site, and re-added it. Same problem. Tried permissions repair, and running ONYX - cleaning everything. No joy. What could be the problem? Other computer on the same account works ok.
ALthough you removed and reinstalled the app, it may have left settings around - either in a dotfile (unix style), or in your Library directory.
What you could do is perform the uninstall. Then from a terminal:
You should delete all the results, and then try to reinstall it.