I tried the scenario on King. I thought I was doing OK but i ended up failing miserably. Defeating the city states seemed like a total waste of time.
The problem is, from when they implemented Rise of the Mongols, they made -Horsemen weaker -Horsemen do significantly less damage to cities -Cities heal twice as fast with walls -Cities are hard as balls to capture
This means that you can't rely on Keshiks and Horsemen to do the scenario. You pretty much need iron units to do it. Securing iron first seems like the highest priority.
I may be wrong, but it seemed like a full health horseman would pretty much suicide against a 1hp city. And defeating a citystate for a reward of 3 horsemen wasn't all too helpful. Maybe there is a real solid strategy to doing it but when you have to travel the world with nonhorse units things just become rapidly unfun.
I think Time is most important thing in Mongol scenario.
Because Mongol really lack tech., if you don't act fast, then others civ will have advanced units that completely own your Keshiks or at least the city's HP will too high to take down with Keshiks in late turns.
and as city-state, I find out that to take down Western Xia and 2 city-states at west (1 is culture, 1 is Horse units ) is very good for begining.
I first take down culture one, that made upgrade cost half to upgrade all my horse units to Keshiks (keep 1 to take down city) *remember before taking the city itself, you need to wait to have your own policy upgrade (x1.5 XP first, otherwise if you have 2 citys your culture points for policy upgrade will be highter.)
then take down Western Xia and use 4 Keshiks and 1 Horses fight in China. (keep Caital to produce horses and upgrade them to Keshiks) *DON'T use horse to damage the city as it will easily lose the horse. Just use the Horse to take the city.
and other units keep go to west (Persia ) and take down the Horse city-states in their way.
you will have 5 units to fight China, and about 5 units fight Persia. (with each term with 1 Khan of course.)
another important thing is Happinese, but it is ok, if you have extra money then just annex a city and buy a courthouse.
and about iron, I really think it is not much useful as it is really slow for a big map. I perfer promote Keshiks to have vs. city bonus.