Final answer:
Option (A), The Chagatai Khanate, under Khan Duwa, tried to conquer the Delhi Sultanate but failed. It was the Mughal Empire, founded by Babur, that eventually established itself, with ancestral links to the Mongols.
Step-by-step explanation:
The leader of the Mongols when they attacked Delhi was not Genghis Khan but one of his descendants. The specific khanate that tried and failed multiple times to conquer the Delhi Sultanate was the Chagatai Khanate. Khan Duwa of Chagatai made the first of several attempts not just to pillage but to conquer at least parts of the Delhi Sultanate in the 1290s.
However, these attempts were repelled by one of the sultanate's most capable rulers, Alauddin Khilji. The Delhi Sultanate was eventually overthrown, leading to the establishment of another dynasty with lineage connected to the Mongols - the Mughal Empire, founded by Babur, a descendant of Genghis Khan and Timur.