You connection string should look like...
Data Source=.\SQLEXPRESS;Database=<your database name>;Integrated Security=SSPI
The error you're getting is because VS generated a User Instance connection string.
A User Instance is a separate instance of SQL Express that is started at runtime which tries to attach the database specified for use by the application. Your database is already attached to the parent instance of SQL Express, and databases can only be attached to a single instance at a time.
By changing the connection string, you'll be using the parent instance, where the database is already attached, rather than the User Instance.