After trying everything I could think of multiple times without success, I ran across this:
http://wangz.itysc.com/2012/08/nuget-install-package-remote-server.html
Tony says he had a similar problem with VS2010 (and I do have 2010 installed) so, out of desperation I installed that Service Pack (which doesn't appear to be related) and it magically fixed it in VS2012!