How to access SQL Server 2012 on EC2 server from my ASP.NET website published on the same EC2 server?
Change the data source parameter of the connection string to "localhost".  When everything is running on the same machine there's no need to specify the actual host name.  If you do need to specify the host name, when SQL Server is running on one server and the database is on another server for example, only specify the name of the server.  In your case the connection string would reference the internal name of the server.  In that case the connection string would look something like this.

connectionString="Data Source=ec2-46-137-228-121;Initial Catalog=MyDb;User ID=something;Password=something"


What ever the value in the data source you should be able to ping that name to make it work.