You're going to need to do a couple things here.
The bottom of your sources/list is all messed up and for intents and purposes will need several of the end entries at the end removed.
Firstly, though, let's get the partner repos enabled.
In a terminal run this
Enter your password
gksu gedit /etc/apt/sources.list
A text editor window will open with the sources.list file.
go down to the entry
and remove the # symbol from the lines
Then while still in the gedit window go down to the last section you have and delete everything after
So remove these lines
The lines you added are wrong anyway.
When you finish this save and exit gedit.
Then while the terminal is still open, run
If no errors appear, then try installing skype again.
sudo apt-get update
If errors do appear, post them.