I would first try creating an Entity Data Model (.edmx file) from an existing database and then create an new (empty) database within VS2013 using Server Explorer if you haven't already. Then use the Data Entity Model Wizard to create new tables/schema from a working database to your empty database using migrations. If this doesn't work then you know your problem probably exists with your SQL Express Instance and not how you configured your connection strings for this project.
I tried these steps in an MVC 4 project in VS2013 and had no problem so maybe you can try also:
STEP 1: Create Date Entity Model from existing db. Right-click Solution Name -> Add New Item -> ADO.NET Entity Data Model
Note: Uncheck save connection string because we just want the sample ADO.NET Model when using existing database
STEP 2: Create new Database from Server Explorer -> Data Connections -> Create New SQL Server Database (test connection)
STEP 3: Right-click on model in .edmx diagram and select Generate Database from Model...
STEP 4: Enable-Migrations, Add-Migration MigrationName, update-database from NuGet Console Manager.
NOTE: You will have two connection strings if you plan on using Authentication out-of-the-box. I also installed EF 6.0 from NuGet.