I tried installing googleearth several months ago on Ubuntu 10.10 using Synaptic and was not successful. Used the following commands to successfully install it. Since Mint 10 is based on Ubuntu 10.10, they might work.

Verify that lsb-core files installed in Synaptic. In my case they were not so I did: sudo apt-get install lsb-core
You indicate above these files are installed so ignore this step.

Next run the following command:
sudo apt-get install googleearth-package
Then enter the following to build the package:
sudo make-googleearth-package --force

You will probably see a number of error messages. Ignore them. Run the command: ls -l /home/user to see if you have a googleearth file. Replace "user" above with whatever your actual user name is. The next step is to run the command:

sudo apt-get install gdebi-core
which is supposed to assure dependencies are installed.

If you have a googleearth file in your /home/user directory, change to that directory in the terminal and run this command:

sudo gdebi googleearth_6.0.1.2032+0.5.7-1_i386.deb
Verify that the file name is exactly the same, you will probably have a newer version than above.

When this is finished, go to your main menu and internet applications and you should have googleearth there. This worked on Ubuntu and it may be different on Mint so can't promise success. I don't have Mint to test it on. Good Luck.