Windows 7 laptops cannot install IBM i Access service packs

For a couple of years now I've had trouble (read unable) with Win 7 32-bit
laptops when trying to install IBM i Access 7.1 service packs. I get this
error:

"This installation package could not be opened. Verify that the package
exists and that you can access it, or contact the application vendor to
verify that this is a valid Windows Installer package."

I decided it's time to figure this out. I found this page:

http://www-03.ibm.com/systems/power/software/i/access/windows_sp.html#action

which in turn sends me here:

http://www.microsoft.com/en-us/download/confirmation.aspx?id=26347

I have downloaded vcredist_x86 (they're 32-bit laptops), run it, rebooted,
and it did not help. Same error.

What am I missing?