What is the problem?
First of all my configuration is this:  I'm developing on a Windows 7 machine running IIS and PHP installed (using FastCGI - It works great!).  I am using PHP 5.3.  I am also running XAMPP for mysql as I don't like to install to many database systems on my PC and XAMPP is portable - you just turn it on when you need it.

I Often take snapshots of my sites in order to work on them.  I do this using Akeeba backup for Joomla and then download the file.  I run the kickstart.php page to then install the site locally.  When it gets to the database information, no matter what I try with the standard settings it fails to connect to MySQL.  My local sites also have recently failed to connect to MySQL.  Through all this I have not changed any of MySQL's settings.  XAMPP can also connect perfectly fine.  So the change must lie with IIS or PHP which have both been recently updated.

What is the solution?
When asked for the host, during the Akeeba install or any connection prompt for that matter, most people use "localhost".  I have for years.  However this fails.  You need to use your localhost BUT type the IP address in its place: 127.0.0.1