In this passage, rousseau refers to political inequality. of the inequality listed here, which enabled the old regime to maintain for as long as it did?

You did not attach the discourse, but I assume it to be the Discourse on the Origin and Basis of Inequality Among Men (the original titile in French is: "Discours sur l'origine et les fondements de l'inégalité parmi les hommes"), also know as "Second Discourse". In this discourse, Jean-Jacques Rousseau emphasizes that private property is the basis of all inequalities.

Therefore, what enabled the old regime to maintain as long as it did was private property.