The LSAC will not tell you if you are complete. They will provide other information:
Click on the Applications tab.
Scroll to the bottom of the page and click on "Application Status" link.
This will bring you to the application status, where you can see where you application is at.
The final stage that LSAC can monitor is here. Next to your applications, you want to see the word "Transmitted" in green.
If you click on your school name, it will bring you to a slightly more detailed page. From here, you can click on the application name (for instance, 'Spring 2008') and see exactly what was transmitted. You can see the actual application, etc.
If you don't see "transmitted", nothing has happened. If it's been a few days and this doesn't change, call the LSAC support folks.