I don't use Ubuntu on a daily basis but it looks like in the top panel of Ubuntu Software Center, you are viewing INSTALLED software (see the middle icon, which reads "installed"). Am I misunderstanding the interface?
Is it possible that Kate is already installed?
Try hitting the Windows or Ubuntu key of your keyboard to bring up the software launcher, and type kate and see if something magical happens.
If not, go to the topmost menu bar of the Ubuntu Software Center (I mean very top bar of the screen, while Ubuntu SOftware Center is open) and a menu will appear. Go to Edit > Software Sources and make sure all the sources of software are activated. and then try again to install.