LocalSqlServer was not found in the applications configuration or the connection string is empty

The LocalSqlServer connection string is defined in your Machine.config.

If you don't have a default Machine.config file, it might have been removed. You would then need to re-add it inside your own Web.config.

My LocalSqlServer:

<add name="LocalSqlServer" connectionString="data source=.\SQLEXPRESS;Integrated Security=SSPI;AttachDBFilename=|DataDirectory|aspnetdb.mdf;User Instance=true" providerName="System.Data.SqlClient"/>

You can find your machine.config here:

C:\Windows\Microsoft.NET\Framework\[FRAMEWORK VERSION]\CONFIG\machine.config