I got the same error 1920. I triedto use My Computer -> Manage to manually restart the service. That did not work. I tried to re-install the service on a several different ports that were not 2002 and that were definitely not in use (netstat -anb). I uninstalled my AVG antivirus software. I uninstalled my Spybot Search and Destroy antispyware software. None of these worked either. I double checked my firewall settings on my local PC, on my DSL Modem, on my other routers. Again, nothing worked. The service would simply not start up, and I could not find the reason why. I even resorted to completely uninstalling LogMeIn, and even going into regedit to delete all the HKEY_LOCALMACHINE settings and entries for LMI, LMI Systray, and so on, and then re-installing LogMeIn. Again, none of thise worked. The service would simply refuse to restart for no apparent reason. I even looked at the LMI log files to try to see what was going wrong. Again, nothing worked. I also used Task Manager, and systematically killed off each process, one by one to see what other software could be interfering with LogMeIn.exe. Again, nothing worked.
Ultimately, the one thing that did work was when I ran Windows XP Pro Update tool, and upgraded to WinXP SP3. After installing that new upgrade (for free), LogMeIn automatically started up just fine. That was the trick. Nothing else described above worked.
So if you're in no man's land after reading all of the threads here, and are just about to throw LogMeIn into the trash, and no longer use it, trying upgrading to at least Windows XP SP3. That did the trick for me where none of the other tricks worked.