To start with, they did ‘annex’ a big part of Europe from Ural Mountains and all the way to Carpathians and Danube. Turkey at that did not exist: there was a falling apart former Seljuk empire with no possessions in Europe. Baltic States, Benelux, etc. are anachronistic names most of which do not make sense within framework of the mid-XIII century other than purely geographic references.
Now, about conquest itself. An answer is simple: they could not without a major change of the pattern of their conquests and some other fundamental things. They would need completely abandon conquest of China, stop activities on the Middle East, move capital to the steppes of Kazakhstan/Western Siberia and then to Volga and, during the reign of Genghis, to abolish a traditional system of apanages. Combination of these actions would allow a concentrated effort in the Western direction. Within the OTL arrangements there simply were not resources adequate for the task. Of course, conquest would have to be done intelligently, by attracting the local resources and offering reasonable conditions to those ready to became vassals (as was done in China, Rus and elsewhere).
Of course, even then ‘annexation’ would mean vassal relations toward the Khanate as was general pattern in the cases of non-steppe territories (after Genghis) because it would be rather difficult to maintain a large scale nomadic presence outside the steppe belt. The whole thing could somewhat change over time with a gradual change toward sedentary/semi-nomadic life style.
To stabilize control over these possessions there would be a need for conversion into Christianity, preferably Catholicism, creating a modified HRE ruled by the Genghizides. Conquest of the Latin Empire and overlordship of the restored Byzantine Empire would lend a new regime an additional prestige because a new Emperor would be overlord of the whole “Roman Empire” with the rest of the European rulers being his vassals.
There would be also a need to revise Genghis’ Yasa to expand it to nin-nomadic people.
All this being said, why would anybody marginally sane chose conquest of a stinking dump of the XIII century Western Europe over conquest of China and Middle East?