I remembered the passage you were thinking of and I did a little digging; I'm pretty sure you're thinking of the Weilue, a 3rd century Chinese text that included information about China itself as well as its relations with other empires. Yu Huan, the author of the Weilue, didn't travel to Rome himself but instead collected information from older Chinese sources along with news and information he could gather from sailors and other travellers.

There are several sections of the Weilue that concern Daqin (the Chinese name for Rome;) The passages about juggling and democracy are included below from Section 11:

This country (the Roman Empire) has more than four hundred smaller cities and towns. It extends several thousand li in all directions. The king has his capital (that is, the city of Rome) close to the mouth of a river (the Tiber). The outer walls of the city are made of stone.

This region has pine trees, cypress, sophora, catalpa, bamboo, reeds, poplars, willows, parasol trees, and all sorts of plants. The people cultivate the five grains [traditionally: rice, glutinous and non-glutinous millet, wheat and beans], and they raise horses, mules, donkeys, camels and silkworms. (They have) a tradition of amazing conjuring. They can produce fire from their mouths, bind and then free themselves, and juggle twelve balls with extraordinary skill.

The ruler of this country is not permanent. When disasters result from unusual phenomena, they unceremoniously replace him, installing a virtuous man as king, and release the old king, who does not dare show resentment.

I should note that the John Hill translation of the Weilue I linked suggests that the story about them replacing the king after a disaster may have been simply a fanciful retelling of a "tall tale" about an idealized distant country.

FYI, I'm not a historian, just a reader, and this is my first submission to AskHistorians so please be gentle!

The Mongol Empire was founded in 1206AD, (Unified Rome fell in the 500s AD) if the Chinese delegation went to Rome to discuss a nomadic threat it could have been any number of other historically powerful nomadic empires including the Xiongnu, The Rouran Khaganate, the Go-Turk Khaganate, the Huns, and several others.