The link icon will only be active when you have highlighted some text in the editor. I've noticed that once in a while, if I highlight some text, and wordpress does an autosave between my highlighting the text and then clicking the link button, but the link button will be inactive. I just highlight the text again and it becomes active.
Typically when there are issues in the dashboard, logging out, clearing your browser cache and cookies, restarting your browser and then trying again will solve the issue.
What browser and browser version are you using?