Thanks for the suggestions.
@dobarton: I think I did see your response in the other thread, but then couldn't find it again. That sounds like something I can try as the next step after the no-arm drill I described. Whether I can progress to coordinating the kick with the stroke remains to be seen.
@CoachFlppr: I'll try that, too. I am curious how long I can keep horizontal balance without kicking. If my legs start sinking, do you recommend just stopping and restarting (like I'd do with Superman glide), or temporarily kicking lightly to regain balance? With the former, I'll probably have to overcome frustration, but with the latter, I am afraid I'd kick the wrong leg again.
I'll experiment with both approaches tomorrow morning.