Thanks for the picture and info.
I lucked out. The the nut on the inside/underside had enough bite that I could just turn the base and tighten it up. As loose as it was I was kind of surprised.
The company that used to make the K-40 (and K-30) antennae, and who still comes up on Google, sold the company. Another outfit makes them now. They have zero info on their web site but their tech support is good. I e-mail them a question and got a reply within an hour. They get points for that if not for having detailed shcematics on their web site.