A surviving Kievan Rus would be interesting. I also wonder what effect this would have on Poland, Lithuania, Hungary, the Balkans and Anatolia. The Byzantines were already in trouble, but the Ottomans were not neccessarily destined to win out against other rivals. Hungary would remain stronger. What would the effect be of this? I assume that they would not end up under the Habsburgs (they were still not particularly powerful at this time). No Mongol Empire would have a profound effect on the whole of Eurasia.
Kievan Rus was “dead” by the end of the XII century due to the joined efforts of Polovtsy and Central Russia: Great Princedom of Vladimir became the leadIng principality with Kiev demoted to the second place. People had been fleeing from the Southern lands exposed to the nomadic raids to the more secure regions and it looks like a transit traffic by the Dnieper lost most of its importance.
What could be different without the Mongols (destruction related to the conquest is quite obvious factor but it did not necessarily played a major role in what followed):
1. Traditional system of passing the throne to the most senior member of the family with the following reshuffling of the lesser thrones could survive for a longer period delaying effective consolidation. OTOH, in OTL for a while the Mongols had been giving throne of Vladimir in an arbitrary way based on the bribes and personal sympathies so perhaps a timeline of the father to son succession would not be too different. Of course, elevation of the princes of Moskow into the Great Pronces of Vladimir was anything but inevitable.
2. Without a credit received in a process of fighting against the
Muslim Golden Horde, Russian Orthodox Church may not get all its OTL influence making the Russian state more open to the contacts with the Catholic “West” and an absence of the self inflicted cultural isolation could produce a significant change.
3. With Vladimir Rus being more oriented toward Volga, expansion along the river and to the East of it is more or less to be expected, which means a clash with Volga Bulgaria. Most probably, even if just by the virtue of the numbers, Bulgaria would be on a losing side. Of course one can imagine a touchy picture of a peaceful co-existence but I would not bet on it. Without the Tatar factor expansion down the Volga could happen much earlier, subject to the timetable of creation of a reasonably unified Russian state. Orientation toward Volga means a trade over Caspian Sea with whatever state is on the other side.
4. Expansion of Lithuania in the OTL scope would be dependent upon survival of the Princedom of Galitz. In OTL it was seriously weakened by the Mongolian invasion and the following raids so, minus that factor, a lot would depend upon the internal stability of the regime(s). Involvement of the Vladimir Rus is possible but unlikely because at that period the Vladimir Rus is busy with a consolidation process. However, Lithuanian possession of Kiev and Southern Rus could be important enough for the Princes of Vladimir to interfere. How effective such an interference could be is anybody’s guess.
5. Absence of the Golden Horde and then Khanate of the Crimea would make a lot of a difference both for Russia and Lithuania (if it ends up with the OTL borders). By the time of the Mongolian invasion Polovtsy were almost contained and probably were not getting stronger. So the whole steppe factor, which existed well into the XVIII, is not quite there and it is even possible that there would be an earlier encroachment into the steppe lands.
6. The Ottomans, if they or their equivalent happened, are not getting the Crimeans in their armies, which could make a noticeable difference in their wars.
7. Probably an earlier fall of the Outremer.
8. Continued fighting between Khwaresm and Baghdad Caliphate with an unclear outcome.
9. There is a possibility of the Vladimir Rus pushing Westward toward the Baltic coast starting with subduing Novgorod and Pskov and then pushing into Livonia (Estonia). Chances for success would greatly depend upon the timing, other players and strength of the AH Russian military system.