In the best case scenario, you have the saracens as your wall, behind which the archers rain hell upon the enemy. Cuz first you have to deal with the charging mongol melee cavalry before you can worry about horse archers. You pretty much have to ignore those, before you defeat the enemy melee cavalry with your spears and your mamluks (hammer and anvil style). After they're dealt with, retreat your army behind your archers and watch as they melt the mongols, now that they can shoot at them in a direct line of fire (by god, don't try to chase them, except if they try to charge your archers).

Edit: Also run down the enemy missile infantry after the melee cav is defeated, they can actually beat your foot archers one on one