OK, when you boot after the "desktop" has loaded press Ctrl-Alt-F1 on your keyboard which should take you to a console (TTY1). Login there and type:
sudo apt-get install gdm3
It will ask for your password, which is OK. Let it finish and then reboot:
sudo shutdown -r now
I can't believe that gdm3 is missing from synaptic. It's an integral part of your Mint desktop environment. If there are any errors write them down exactly and post back.
*** NOTE - I'm assuming you are using the GNOME desktop that Mint comes with. If not, post back with details before you do the above ***