Thanks for the advice, guys. There's one more way to get horse archers (although it's not exactly reliable): Jihad! They often contain Mamluk HAs for some reason.

I finally got the Sinai built up for Mamluk HAs in 1146. I'm quite happy with them - their morale is good enough to run round the back of the enemy without routing, and they are sort of OK in melee - i.e. good enough to kill archers.

Now I've got the castle-level horse archers, it's time to think about the Kwarazmiam cavalry: citadel-level knights...