Question

even though they were highly influenced by Egyptian culture, in what ways were the kushites able to hold on to their own political culture

Answer 1

Here this might help you....
Even though they were influenced by Egypt, this country had its own culture. They still had strong rulers who were females. They also had their own way of making pyramids

and this to....
 The Kushites controlled the army once they had conquered Egypt and the Kushite king Piankhy became Pharaoh; the Kushites always provided the major portion of the army during a subsequent series of wars with Assyria for the control Syria. 

The Kushites were a warrior society, so that's how their soldiers were organized and fought, whereas Egyptian soldiers were professional regulars, a standing army. 

Egyptian military commanders were basically Kushite puppets, so there was some strain in the relationship between the Kushite part of the army and the essentially subject Egyptian contingent, as the Kushites always made sure they had the upper hand during the 90 years or so of Kushite rule.