It seems as if everyone in the college has been fighting Ancano for about a week but he won't die. His health is down to about 25% but no matter what we throw at him, it won't go any lower. I can't use any weapons or shouts on him either, although he is being quite free with his fists.

This was getting ridiculous. I tried a console cheat to advance the action to the next stage, then I went away for a while to get some supplies and when I came back, everyone was telling me I was suspended for my actions. Huh! Now I am greeted by Quaranir as the Archmage, but the fight in the hall of elements is still going on. I know there are some bugs in this stage, but I don't know what to do.
Should I try the whole stage "Defeat Ancano" again ?

Yeah do it over, have you tried using the Staff of Magnus on the Eye when they ask you to?

Yes, I did that, but it was a bit confusing. I'm going to start it again. Thanks