A New Challenger Appeared!
(You're all gonna hate me for another seemingly impossible voyage, but it'll help to know that it isn't a government-funded project and the King has no idea what they're up to.)
India - 1424:
While the seemingly endless stream of wars in the subcontinent continue, with the Vijayangar-Reddy alliance fighting against the northern Bahmani sultanate, the news of the Chinese discovery was hitting ports across the world. Tired of the endless and (to them) futile territorial struggles in the Indian subcontinent, a group of heavily religious Hinduists begin organizing an expedition to the New World, to get away from the useless wars. By December, they've amassed 100 ships ranging from fishing boats to AWOL war ships, containing farmers, fishermen, sailors, women, children, cows, soldiers, workers and slaves (who've run away from their masters and are told that they will be granted freedom in the New World.) In total, the number of people there is around 2,500.
(A map of their voyage this time

)
(See? Not totally impossible.)
Of course, with a voyage going that far south, into uncharted waters, there were some difficulties. Many of the children on board, for example, came down with a cold, and ended up freezing to death. One fishing boat, mostly containing high-ranking soldiers and worried mothers with their children demanded to head back, and were lost among the waters.
They reached South America in August 1426, landing on the shores of OTL Northern Chile (ignore the map, I didn't show accurately where they landed.) At first, they land and unload the boats, gazing in wonder at the high mountains in the distance and strange surroundings, but soon got to setting up a shelter. They had soon erected a couple huts atop a nearby hill, and when night fell, there were enough to house a quarter of the people. The others slept on the ships.
In the morning, the religious among them prayed, while the workers and soldiers started construction of a wall around the hill on which their huts stood. They set down a foundation of wood and crates brought on the boats, and then split into two groups, the one to finish the habitations, the other to begin carving stone bricks out of the hills and cliffs around them. This work continued, and by the end of 1426, they'd finished one half of the wall surrounding their thatch-roofed huts.
It was in May of 1427, when they'd just finished the wall and were having a celebration in the middle of their ramshackle town that a small group of twenty or so men appeared in the distance. They didn't speak any recognizable language, and had a strange appearance, but it was clear that they were as shocked as the Indians to find there were people there. They confusedly exchanged gifts, seeds (of maize) from the Aymarans (for that's who they were) in exchange for a cow from the Indians. The Aymarans went away shocked while the Indians were shaken by the revelation of another group of people nearby.
June of 1427: a baby is born to an Indian woman. She dies in childbirth and the baby dies soon from undernourishment and an exotic disease.
By 1428, they'd engaged in communication with the Aymaran natives, who came from a city just up the coast from their settlement, and soon established a rudimentary form of communication. They had traded some more livestock for native plants and native animals, and had also given the Aymarans gunpowder weapons and iron blades, in exchange for help in building stone houses and another wall around the hill. With the Aymaran help (which was great, given the rush to see the newcomers from the civilization), the Indian hill-city was completed, with 50 houses, a mandir, and gates in both walls, was completed finally in January 1429.
By this time, not only had the Aymarans developed a healthy relationship with the Indians, several other groups including the Chimu, Chiribaya and some merchants from the Kingdom of Cuzco had come to visit and trade with them. Some Indians took wives of Aymaran or Chimu women, and they went and lived in the City.
Healthy trade and good relations continued among the Indians and the Natives for many years.