UPDATE:  I finally found an answer here:  LifeCycle Designer ES2 - Does Not Recognize MS ... | Adobe Community.  The trick is to remember the final "\" at the end of the registry key you have to add, and then to reboot.  Now Livecycle will find Word 2010.

I have the exact same issue.  I've tried installed SP1, SP2, and SP2.1 "quick fix", but they all give me a Windows Installer Error message saying "The upgrade patch cannot be installed by the Windows Installer service because the program to be upgraded may be missing, or the upgrade patch may update...