Yesterday I was, for the first time, able to incorporate kicking in sync with the whole stroke, ending the day with finally being able to keep it in sync for the length of the pool (25yd). Today I swam many lengths (didn't count, but over 20 for sure) and only lost rhythm 2 or 3 times. This is a major breakthrough for me, and I am pleasantly surprised that I got it only a week after discovering that I had a problem with always kicking the opposite leg.
Swimming felt better than ever today, even though I am almost sure that I was doing most things worse than normally (because I was focusing exclusively on my kick and hip movement). But with my core rotating as a unit (more or less - obviously this is still very far from good skill) rather than twisting, movement felt incredibly easy. And doing a 2BK instead of 6BK probably helps with that ease, too (although I am pretty sure that's less important than kicking with the rotation instead of opposing the rotation). I also feel that 2BK helps with keeping legs streamlined.
I first succeeded in kicking in sync by starting from a Superman glide, beginning to move one arm, and kicking the same leg as I started the pull, then waiting to kick the other leg as I got to the same arm position on the other side. Initially it was definitely helpful to think in terms of "pull and kick with the same leg"; today, I was also able to change perspective to "reach and kick with the opposite leg". Now I can think of it either way and it works.
Thanks for all the advice - I followed most of it to help break the bad habit, and will continue to do the recommended drills. But being able to also swim whole stroke again makes swimming more fun.