Catch 22 cannot install windows 7.1 SDK because cannot reinstall Visual C++ 2010 SP1 Compiler Update for Windows SDK

I'm running Windows 7 64-bit SP1. Some time back I had the 32-bit Visual Studio Express 2010 installed. Then I attempted to install the Windows SDK 7.1 and got a failed installation. So I uninstalled VS 2010, and tried again, including getting Visual C++ 2010 SP1 Compiler Update for Windows SDK installed, but there was still an installation problem. I look and find an installation is acknowledged by the Programs & Features control panel, despite the earlier failure. I run MicrosoftFixIt and uninstall that failed installation, and clear out any files & folders left behind.

Now to the present. I run the installer for SDK 7.1 again, and it halts and complains. Exactly as here: Cannot install Windows SDK 7 and .NET Framework 4: Unknown product

So I follow the advice and attempt to install the update mentioned, and this is blocked because Windows SDK 7.1 is not installed, of course, because it's a prerequisite.

I'd like to clean up the mess and install SDK 7.1, but I can find no way to do that. How should I proceed?