I had some Connection Broken issues today, though it did not work and then break as you're saying. The game patched for me this morning after which it was always a broken connection... but I noticed that the date on the client and the date in the launcher for the latest client download were not the same, so I did the "Verify and Fix" option, let it run, and then restarted the client and everything worked.
Maybe start with that? Let it verify that the client has no file errors.