Download Client appears then vanishes
It’s running in the background, but for whatever reason hasn’t finished loading. Perhaps try heading to https://account.arena.net/content and see if you can download a new client? I highly recommend the 64-bit version, which wasn’t available at release.
[Edit: If it’s still running and you need to restart it, you can end the task from Task Manager, which you can access via Ctrl-Alt-Del. But it sounds like the client just isn’t working for you, so hopefully a new one might fix it.]