It seems in 1206, Genghis khan had around 120.000 soldiers. Later on he included more foreign peoples in his army, uyghurs, manchus, defected chinese commanders coming with their units, peasants used as meat shields...
50 years later hulegu arrives in persia with an army numbering around 180.000, supposedly gathered by taking one in every ten soldiers of the mongol empire. Since mongol population wouldnt be able to rise so quickly, I'm guessing most of these troops were of non-mongol and sedentary origin. Or maybe mostly mongols with auxiliaries like chinese engineers? How much of the mongol army was actually mongol?
I think nobody will be able to reliably answer this question. I have wanted the same question answered as well.
We have basically no writing by step people with few exceptions, specially not unless they already have created some political entity.
There are a couple ways we could try understand what 'mongol' even means.
Are we talking about an ethnicity? Or are we talking about a language group? Is it a religious entity maybe? A way of life? Depending on your interpretation this changes your whole analysis.
There was a lot of ethnic mixing and we have essentially little understanding where a hun or a turk begins and a mongol ends based on ethnicity. Were Turkish speakers that still lived in the Eastern steppe ethnically more like mongols, or more like the Turks of the middle east? Nobody can answer this questions.
We have the same problem with regards to language. Language is very dynamic and we have very little understanding beyond large scale language transitions, about the distribution of languages on the step and how they evolve. Does the language we now call 'mongol' refer to the language of just a few ethnically mongol tribes with many other dialects and even languages spoken by other groups with the same ethnicity?
The next problem is assimilation. Genghis would defeat tribes and simply absorb them, mostly by taking the woman and children but also add his army. Now some of those tribes we would probably consider 'mongol' other we might call 'turkis' but again, those are language based differences. So if a Turkish speaking tribe is adsorb into the mongol nation, are they mongols?
And if we went back and asked people back then, they might see the difference between tribes that we simply call 'mongol' as more relevant then the difference between a mongol and a turkish speaking tribe based on things we don't understand. Our understanding of steppe politics in a non-imperial system is very badly understood.
So really you have to make assumptions about how much of the steppe people in say 1206 you would consider mongolian. And then do some rough geographic population estimation and so on.
I'm not gone attempt to give a prefect number, in my mind I usually conceptualize it as 1/3 to 1/5 of the steppe army (not considering engineers and so on) as mongol depending on the time period.
For understanding all the tribes and so on, you might want to read:
While I don't agree with everything in this book, it does give some context about the steppe:
The problem is most books about the mongols, are simply about military conquest.
Not a book but something that contains a lot of information is the set of lectures (unfortunately not free, but I don't know of a comparable free resource) by Professor Kenneth W. Harl, Ph.D of Tulane University. Again, he does not attempted to answer the question directly, but the information is as new, up to date and modern as you are gone find anywhere. Better then many of the books on the subject.
https://www.thegreatcourses.com/courses/the-barbarian-empires-of-the-steppes.html
Maybe some Ph.D. out-there has or is writing a analysis of steppe populations and ethnicity but as far as I know, I have not seen it anywhere.