What is the name of the Mughal ruler who constructed a mosque within the compound of Khwaja Muinuddin's Dargah?

Akbar

The correct answer is Option 4 - Akbar 

Akbar, the Mughal ruler constructed a mosque within the compound of Khwaja Muinuddin's Dargah.

Akbar went to Khwaja Muinuddin's Dargah fourteen times, sometimes two or three times a year, to seek blessings for new conquests, fulfillment of vows, and the birth of sons. He maintained this tradition until 1580. Each of these visits was celebrated by generous gifts, which were recorded in imperial documents. For example, in 1568 he offered a huge cauldron (degh) to facilitate cooking for pilgrims. He also had a mosque constructed within the compound of the dargah.