What is the best way to get nut-grass out of Monkey Grass?

You cannot use anything.You will have to dig it or smother it with newspapers and cardboard

Pull it out !

elbow grease works well -- nut sedge seems to LOVE mondo grass!!

This suggestion comes from a landscaper, and I've had good results using this method. "Paint" the nut grass, with Roundup. Using a cheap thin-tipped artist's brush, paint each petal of the unwanted grass sparingly with a small amount of the Roundup being careful not to as little as humanly possible on any of the grass you want to keep. It doesn't take much because the Roundup is systemic, meaning just a little will kill the entire plant. This is time consuming and tedious, but effective . I've had great success ridding my thyme groundcover of oxalis with minimal damage to the groundcover.

There is a special herbicide you can purchase at a turf store or irrigation company, sorry cannot recall the name of it, however our friend 'the landscape architect' recommended it....it worked, the only thing that worked. He called the grass 'nut sage' - has a nut shaped root and triangular blade. It will come back without this treatment.