Well, Maybe I could sum my point of view in few points for- and against- Mongol victory in Europe.
The points in favour of Mongols:
1. Quality. The Mongol army was without a doubt most effective war machinery of the time, leaving european armies with their terrible (when deployed against steppe style) fighting strategy in great disadvantage.
2. Politic situation in Europe. European Leaders of that time seemed not to care and even realize the seriousness of situation, and there was no unity, with guelphs and ghibellines in Italy and Anglo-French conflicts, which means they would probably not be able to put together army strong enough to face the invading Mongols. Also, the morale of such army might have been weakened by horrific reputation of Mongols.
for additional sources see Wordsworth Reference books "Dictionary of Military Biography"/Subatai Ba'adur
But, what is very interesting, when the Mongol horses reached the azure Adriatic, Frederick II, Emperor of the Holy Roman Empire and King of Sicily, expressed the opinion that it would be good to use them as allies in the struggle against the Papal throne; but the idea was forgotten in 1241 when the Mongols left. It would be interesting to see what would happen if they stayed. They were capable of diplomacy as well.
The points against Mongol victory:
1. Numbers. Only maximum of 2 or 3 tumens (20,000-30,000 soldiers) participated in European campaign, so no any large numbers of Mongols against most of western europe.
2. Fortifications. The Mongols had serious problems with attacking fortified towns, and castles respectively.
3. Morale. Problems with fortifications and maybe stay in foreign countries for such a long time (1235+) would probably lead to fall of morale among Mongol troops.
4. Size of the conquered territory. It would be impossible to manage the whole conquered territory with a single army. When Ogedei died in 1241 and power fell into the hands of the queen Toregena, Guyuk's mother, Guyuk's and Bun's followers were recalled, and poor Batu (Leader of Golden Horde armies) was left holding an enormous country with only 4,000 faithful troops and with extremely tense relations with the central government because of previous conflicts with son of Ogadai, Guyuk, which meant no support for Batu. There could be no question of holding the conquered territories by force. Internal conflicts between leaders the of Mongol Empire should be also kept in mind.
My personal opinion is the Mongols would be able to plunder large areas on their way into the Western Europe as they were only week's ride from Paris, but their ability to conquer larger cities at the time is questionable, because they didn't carry their siege equipment
Wrong facts,wrong conclusion.
1) The operation was planned meticulusly about eight years before its execution.The planning was something that von Moltke and his staff would envy.The army allocated for Southern Russia- eastern Europe was 150000.
From 1231 the Mongols were collecting information on political and economic situation in Europe,even about the connections of ruling class families;Subodai,on the strength of that information said that he needed 18 years to conquer Europe down to the Atlantic coast.(see
www.mongolianculture.com/mhistory.html)
2) Mongols din't know much about sieges until they conquered China;there they aquired engineers,materials(gunpowder) and the necessary know how,but the chief planner and field commander was an Alexander-class general and he knew that
is better to dominate your enemy in the open and maintain the initiative instead of losing it in a siege and give your enemy time to recover(Vicompte de Tyrenne).
3)Most of the Mongol cavalry were away from their country conquering foreign lands for years and were used to living on plunder.
Their law,the 'yasak' equated military disobedience with common law crimes.No question of discipline ever arose In Mongol units and their moral was never yet in question.
4)The Mongols did not invade Europe with one "army",or let's say one army as they meant it then in Europe,The mongolian army was divided in four corps which had the strength of an army of that time,for the 1241 campaign.
See :Eric Hildinger's"The Mongol invasion of Europe
www.HistoryNet.com/MilitaryHistory/articles 1997
the Mongol army,for the purposes of the campaign of full winter 1241(a great surprise) was arrayed from North to south in four corps under the command of Kaidu,Baidar,Batu and Kwadan-Subodai placed himself in the central corps with Batu son of Ogodai and nominal head of the campaign,but Subodai was the field commander according to the explicit orders of Ogodai.The army numbered 70000 men as the attack wave,more to follow later when needed.
The Mongols showed a unique sychronicity of the various corps with well defined objectives and time limits per objective.
Their aim was to converge on the feudal army*of king Bella IV and surround it.
First moved the northern arm anihilating the poles and Teutonic knights in Liegnitz;burning the cities in his wake Baidar(senior commander) crosses the Oder in Ratimbor,advances south parallel to the river and defeats the 30000 Germans of Eric of Silesia."The only thing he has to do now is to settle his score with the Hungarians.
Batu and Subotai smashed through the Hungarian border defences and were waiting for the other armies to converge,
whereas Kwadan,after winning three pitched battles forces the passes of the Carpathians and entered the Hungarian plains from the south."The Mongols,after crossing the river Szeis or Sjasso during the knight,surrounded the 100000 strong'
feudal army* of King Bella IV and proceeded to anihilate this force by arrow and incediary fire always retreating before the charging Hungarians and firing.The Hungarian army was destroyed unable to give battle since the Mongols were avoiding it!"(Erick Mauraise:"Introduction to Military History" by Swiss Army Directorate of Military and Historical Studies)
*Eric Mauraise is speaking about the feudal armies with some disdain as in the same book,on middle ages he writes"The achievement of arms were of particularly mediocre value(with possible exception of Bouvin-1214) in comparison to the feats of the Mongol cavalry.That magnificent cavalry had conquered Eurasia from the Pacific Ocean to the Adriatic sea within the space of a generation."
Their superiority was due to:
a)The Mongol horse and the Mongol Bow.The bow had almost double range(*) from the bows of that time,and their hardy ponies from the harsh mongol steppe could endure hardships that no other horse could."they cover distances up to 70 miles a day,moving faster with great ease than our armoured divisions with absolute aytarky since their logistics was minimal(the fastest advance with armour was in 1940 with the 19th panzercorps of general Guderian in his drive for the Channel with 54 klm/day.)
b) their military law 'yasak' which equated military disobedience to the crimes of common criminal law
c) to the vertical organization of their army which was structured in squadrons,regiments and corps of 100,1000 and 10000 horsemen respectively all capable of moving silently in the battlefield with the signs of small battle flags.
Basil H. Liddel-Heart,the formost authority of UK in armour movement based his theory of armour strategy and tactics(combined arms operations with preponderance on tanks) on the movements of that magnificent cavalry;unfortunately his best pupil was not English but general Heinz Guderian,one of the great captains of history since his tanks changed the course of a world war and that of history.
It is of course imperative to know that the Mongol cavalry movements are still studied in all schools of war today and Subodai has given a lot of head and other pains to thousands of aspiring tank officers throughout the world with his ingenius planning and its cloackwork application.