Normally on the focus puzzle there should be beams of light so I know how to adjust everything. However, they aren't visible for me, making the puzzle extremely difficult to complete. Can anyone help me with this?

Have you used the Fire and Ice spells to align the images on the wall?

I've tried, but the problem is I can't see the light beams, so I have no idea what it's actually doing.

This is a problem a number of us have had. For quite a while, I solved it with the the embseries antifreeze. After the latest patch it came back. I was able to get the effects back with another d3d9.dll I found referenced in one of the postings here about missing spell effects. It was at filefront called skyrimbetterperformance. If you search for spell+effect. For that particular puzzle, there I also saw a youtube walkthrough which tells what clicks you need to click if you can't see the beams.

Is this it? http://www.fileplanet.com/223433/220000/fileinfo/Elder-Scrolls-V:-Skyrim---Skyrim-Better-Performance-Mod-v1.1a