Let's say the "greatest extent of the Mongol Empire" would be in 1260, the year after Mongke's death, when the Empire still had not fractured into the 4 successor states. The empire's realm would stretch from Syria to China.
We can examine records of some of the movements various agents took around this time. One must consider the urgency and purpose of movement across great stretches of land in this period. A muslim trader would take a more leisurely path and route than say, a mongol army on campaign.
Then one must also consider the vast differences in terrain across West, Central, and East Asia. The weather and season too, may be a major contributing factor, for example, Mongol armies were known to cross frozen bodies of water during the winter.
Here are some anecdotes I found. Carruthers in Unknown Mongolia claims a Mongol on a single pony can ride from Urga to Kalgan (600 miles) in 9 days. Chinghis Khan in the pursuit of Jalal ad-Din, went from Bamian to Ghazni via Kabul (130 miles) in 2 days. When Batu invaded Hungary, the vanguard when crossing the Carpathians covered some 180 miles in 3 days. In 1259, Khublai left Xingzhou around July and laid siege to Ezhou in September, this was about a 450 mile trip.
I was trying to find concrete sources on Hulagu's travels in 1260. January of that year was when Aleppo fell. We know Hulagu deprated eastwards because of the succession question following Mongke's death the year before. Khublai declared for the throne in May, while Arigh Boke declared in June (Some sources claim September/October? Which doesn't make sense). In Hulagu's absence, the famous mongol defeat in Ain Jalut occurred in September. This compelled Hulagu to return westwards, but where exactly Hulagu was in this time between January and September is something I could not find. This may be bad practice but I'm postulating, based on the readings I've found, Hulagu was in Mongolia, or that was at least his intention in returning home. Whether or not he met Arigh Boke or Khublai in person during their respective councils in Mongolia or China is unclear.
Should Hulagu indeed have been in the homeland, then it's somewhat safe to say his trip from Syria to Mongolia took 1/2 a year to 3/4 a year. Aleppo to Beijing is roughly 4200 miles. Here's some calculations from the anecdotes above.
Smaller groups on pressed speed:
Army groups on campaign speed:
Let's ignore the campaign speed as Khublai would have been inhibited by Chinese armies and various sieges. If we take these calculations as accurate, then 60 miles a day may have been possible should a rider have access to multiple fresh horses. Many of those trips were made over rugged and hard terrain, such as in Afghanistan or the Carpathians. Let's say 50 miles a day to give room for other factors. 4200 miles/50 miles = 84 days. That's about 3 months, on very good speed, riding all day long likely.
References:
I'd look into the last book in particular, it has good info on more "normal" travel conditions.