I think the term "fail" is entirely subjegtive in this context. The Mongols certainly seemed to succede in terms of thier objectives in Eastern Europe.....just as they certainly "failed" to ravage Western Europe.
It certainly is open for conjecture as to what would have happaned had they not experienced political instability. I certainly don't think it was a "forgone conclusion" that they would have penetrated much further into Europe....though they certainly MIGHT have done so.
Heck, I don't even think it was a "forgone" conclusion that they would have taken Vienna. Ancient Warfare has too much uncertainty, too much depends upon the fate of individuals, condition of the battlefield and just plain luck....to predict with certitude the results of a hypothetical battle.
In the game, I think it's best to represent the Invasion by giving the Horde a certain "inertia" factor. As the Horde wins battles it gains inertia. As the Horde takes casualties and pushes into terrain unsuitable for horses it looses inertia. Once the horde reaches a certain "inertia threshold", it ceases it's advance, pulls back some, looses a large amount of strength and goes into defensive mode (i.e. it's main focus of attention has shifted elsewhere).
You could also put in a random check for an event every few years to see if the Hordes leader dies (which would cause a heavy loss of inertia). This would keep players guessing as to how long the Horde would continue attacking even if it wasn't suffering heavly. As for the Horde's millitary, they should be very strong, but still beatable.
This would represent the fear and uncertainty that the Mongols engendered in Europe but still allow the possibility for a capable and dynamic European leader to turn the tide. Which certainly was theoreticaly possible, given the right conditions, combination of troops, leadership and luck.