As drone has said, the bridge battle in Kiev is a good way to stall them, provided you have the right combination of crossbows, halbs, and heavy cav. And, yes, good idea keeping reinforcements in neighbouring provinces. Another thing I've done -- and it led to one of the most spectacular siege battles I ever played -- is fortify Khazar to the max -- upgrading to citadel with full artillery. The mongols are pretty bad at sieges, so a decent garrison of high valour troops should give them a run for their money. As I recall, the battle lasted well over an hour and, while I did eventually lose, the mongol forces were so heavily damaged that they presented only a minor threat thereafter.
Also, your resolve to start in high will make preparations much more difficult. Good luck with that. For me, starting in early, I love being able to prepare slowly and meticulously. Also, Khazar heavy cavalry is only available in early, and I love stock-piling them to go against the horde.
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