Success at last.
This is from a Microsoft document I eventually found.
2. Known Issues
2.1. Installing
2.1.1. Installation or repair of the Windows SDK 7.1 fails after Visual Studio 2010 SP1 is installed
When the Windows SDK 7.1 is installed or repaired after Visual Studio 2010 SP1 is installed, and the SDK installation includes the Visual C++ Compilers feature, the SDK installation may stop and a "fatal error" message may appear.
To resolve this issue:
Run the Windows SDK 7.1 Setup again, clear the Visual C++ Compilers check box, and then complete the setup.
Install this Compiler Update to install the Visual C++ compilers and libraries.
2.1.2. Cannot silently install the Visual C++ 2010 SP1 Compiler Update for the Windows SDK 7.1
This Compiler Update cannot be silently installed.
To resolve this issue:
There is no workaround. Silent installation is not supported.
2.1.3. Recommended Installation Order
To ensure that your system has a supported configuration, uninstall the following products and then reinstall them in the order listed:
Visual Studio 2010
Windows SDK 7.1
Visual Studio 2010 SP1
Visual C++ 2010 SP1 Compiler Update for the Windows SDK 7.1
John