If you have room as far south as Egypt, there's no need to fear the horde. The horde AI is the worst in the game, and immediately after invasion they tend to sit with several stacks of troops in Khazar and send smaller parts of their army in a random pattern across the steppes, which shouldn't be an important part of your economy (especially if you have the incredibly rich Asia Minor). The main danger is if they, as they do very rarely, decide to move a large amount of troops south. The way to tackle such an expedition is to make sure that your fleet is intact and unhindered by enemies, and then let them south. Once they feel secure, kill their eventual vanguard in Georgia, and then crush them when they have nowhere to retreat to. Wash, rince and repeat: the hordes main disadvantages are their inability to mount a proper invasion plan, and their inability to raise new troops. after losing a couple of thousand troops, they tend to be made up of single stacks randomly placed across the plains, which shouldn't be a threat to you.

If you have lands further west, just mount a defense at a river and bring spears and archers. The mongols won't stand a chance as long as you don't autocalc.