I've read around that using the proprietary ati/amd drivers can solve some slowdown and power issues, so I'd like to give them a try. However, if I open the "additional drivers" tab in the Ubuntu software center it says "no additional drivers available" and "no proprietary drivers in use". Why, and how do I get it to check for the relevant drivers?

well you know it is very well documented how to do it the ubuntu way . that documentation has been there since 2004 how to install the correct drivers.
I am sure this will get you to do it the ubuntu way. click thisand it will pretty much talk you through how to down load it and build it the ubuntu way. Ubuntu I think should set there system up a little more user friendly. since they already data mine you. They should spend that money to just have the blob ready to go for you at install time.

!!!!!!!!!!!!!!!!read how to uninstall the one you got the link is there.!!!!!!!!!!!!!!!!

The video device in that machine is the RS880, which is labeled by AMD as legacy device. The legacy driver from AMD does not support the newer kernel and Xserver used in 13.10. Using the proprietary driver is simply not an option for you, unless you go back to 12.04.

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Ah, right, thanks. Wiped and installed LMDE instead - seems to work with no slowdowns.

there is a ppa that has patched that to the 13.04 and are working on a patch for the 13.10 just google it it is there.
http://askubuntu.com/questions/33163...-radeon-hd4250