After a couple of attempts at it we got the Server configured correctly for .Net 4.5 Web API. Having seen a couple of blog posts get this one wrong I decided to post what we found out here. If you are working with Windows 2012 Server you must, must, must use the Server Roles tool. Do not attempt configuration without using this tool, you are only making life harder for yourself.
Windows 2012 server will come with .Net 4.5 installed but you must configure it to run with IIS. Add the Application Development Role and enable ASP.Net 4.5:
Add the Application Server Role:
Enable Features: ASP.Net 4.5:
And of course you must have IIS enabled:
Then allow the Manager to make it all happen (we hum the Jeopardy Television Jingle while this is happening):