I'm probably not the first to ask about this but I need help on figuring out how to fix this and I hope it's not too late. I've watched youtube videos on how the quest is completed and saw that I had already done all that stuff and the quest is still not completed. I even did other quests to point where I'm already the Arch Mage of winterhold and tried going back to speak to the woman you're suppose to and nothing has changed. Whenever I speak to her she just tells me about the problems she was with the other lady. So if anyone has figured out how to complete this please help me out. I'd really appreciate it.