Hello!
I don't know if this answer comes too late but the instructions worked at least for me. After you have deleted all the LogMeIn files and moved them to trash, you have to empty the trash. BUT one of the files couldn't be deleted permanently, so then I went to the Activity Monitor and quitted the process of the file and FASTLY went to the trash and deleted the file permanently. But you have to be fast, because the LogMeIn icon appears again even if you have quitted its process! Hope it works for you!!!!
And here are the instructions:
To manually remove an application and all associated files:
Launch Activity Monitor and change "My Processes" at the top to "All Processes", then make sure the app you want to remove is not running. If it is, quit the process before proceeding.
Launch Finder (see image below) and search your entire hard drive for the app name (hopefully unique, such as Skype)
In Finder, select View > "as List", if you haven't already done so. Do not use "as Icons" for this procedure.
You can narrow the search to specific folders or search your whole Mac, which is preferred
In Snow Leopard and earlier, click "File Name" instead of "Contents" for better results.
In Lion and later, click the + button below the search term, then select "Name" "Contains" and enter the app name. (see image)
Click the + button below the search term to add criteria, then click the search criteria drop-down and select "Other...", then "System files"
Click the "aren't included" and change to "are included"
Sort by name, kind, date, etc. to identify components of the app, such as folders, .plist files, cache files. etc.
Delete all files and folders related to the app.
Don't empty your Trash until you've determined that everything is working OK, in case you need to restore something you deleted by accident.
A reboot might be necessary to completely remove some apps.