Hi Matthew, 

I think you've mixed up 2 separate way of installing dnn. 

in the 1st instance if you have the install version downloaded if you keep the database.mdf file that is in app_data then the connection string you have provided will work... 
<add name="SiteSqlServer" connectionString="Data Source=.\SQLExpress;Integrated Security=True;User Instance=True;AttachDBFilename=|DataDirectory|myDatabaseName.mdf;" providerName="System.Data.SqlClient" />

To Explain the above connection string says that it is going to mount the database each time the application is launched. 
The Other method is to mount the databse in SQLServer, and point the connection string at this db so for example if you mount the database you have created as "MyDNN" the following connection string would work.

<add name="SiteSqlServer" connectionString="Data Source=.\SQLExpress;Integrated Security=True;User Instance=False;Initial Catalog=MyDNN" providerName="System.Data.SqlClient" />

You can find the name that you have mount your database as in SQL Management studio it is the name listed in the Databases folder. The advantage of the second method is you can manipulate the database from with SQL management studio whilst the app is running.

From the current position you have stated you could go either way I'd suggest the second route to do this simple locate the database name replace that in to the connection string above and copy and pase this into the web config file.

Hope that helps and explains it a little

John