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