What the message is telling you is that lightdm is already running in the background. So, login to the console-session/terminal by pressing Alt+F2
, or choose Exit to console login from the window below.

Now, remove MDM, using the commands below:
sudo apt-get purge mdm
sudo dpkg -r mdm mint-mdm-themes
Set LightDM as default from this command:
sudo dpkg-reconfigure lightdm
Then reboot. MDM should be gone by now.
LightDM may start, but it's most likely to be in low graphics mode.

To fix this, you have to edit /etc/lightdm/lightdm.conf
as following:
[SeatDefaults]
user-session=ubuntu
greeter-session=unity-greeter
As you are unable to edit lightdm.conf
through console-session, I may suggest using the Ubuntu Live CD.
(Using the live cd, enter gksu nautilus
on terminal, then locate and open /media/yourUBUNTUpartition/etc/lightdm/lightdm.conf
from Gedit... then you can do the editing!)
Considering that you haven't installed unity-greeter, download the package for Precise from here, and place it on /media/yourUBUNTUpartition/home/YourName
.
That's it! Get out from the live CD and get back on the Console Session. (Alt + F2).
Now install the Unity-Greeter file you downloaded before:
sudo dpkg -i **unity-greeter_0.2.8-0ubuntu1_i386.deb**
Finally start LightDM,
sudo service start lightdm
dpkg -l | grep -i mdm
– ish Sep 09 '12 at 04:17