I had the same problem -- clicking on the Update button in Origin did nothing for me.
Following TuPR's solution, this was what I did:
1. Clean the Origin temporary folders (there are 3 locations on Windows 8). Use the instructions given here: http://help.ea.com/en/article/clearing-your-origin-temporary-files/
2. Restart Origin.exe as Administrator.
3. Origin will ask for a relogin.
4. After login, Origin will verify your DA:I game files automatically (this may take a few minutes).
5. At the end of it, Origin will have installed the latest updates.
