Nutgrass can be defeated if you get rid of regrowth by the time it has 4 to 6 leaves. If left longer, new nutlets begin to develop.

So it's your choice as to how you get rid of top growth: pull it off or apply herbicide. Both methods work. Both will require several years persistence.

It's critical to be persistent. Give it an inch and it will set you back to zero.

I've dealt with lamium intrusions from a neighbor. I susepct your growth is more dense than mine was. The remedy is essentially the same: Remove top growth and repeat.

It might be easier to remove desirable plants to a temporary site, then go after the lamium.

The first step in gaining control of both of these is understanding the type of soil they grow best in. The nut sedges are wetland plants and grow best in moist soils, although they have adapted and will grow in dryer soils as well. Usually the best means of control is to remove all of the nuts that are found in the soil since plant poisons generally do not kill these.
Lamium grows in dry shade, best. The plants like a shady area with well drained soil, but again has adapted to many environments. I have found that a good heavy mulch will control this quite well.

Round-Up doesn't do much to nutsedge. I recommend Sedgehammer, which I used this summer on the nutsedge that was taking over my lawn. It worked very well, but it works best on plants with 3 to 8 leaves (that's taken right from the brochure in the package), so I've had to reapply twice, and I am happy to do so. I sprayed right into a couple of beds, and neither the grass nor any of the nearby perennials showed any effect.

I got the Sedgehammer on Amazon, as no store around here stocks it.

Good luck; it's a dreadful pest.

Thanks to all! Really appreciated!!

Gives me a less panicky approach. Can't lift the plants and move to treat in this situation, but it is late in the season, so there is time to strategize for next spring.

For now, in the 'nut sedge' bed, it has been heavily weeded out, but the nutlets are everywhere. I will try Round-up on any little regrowth this fall, and will look at getting Sedgehammer for the spring .

I will clear out as much of the Lamium, and all the runners I can find, and then hit new growth in the areas I have tried to clear.

And then work on clearing the next patch, and ditto.

One step, then another. But there is so much left to do that I couldn't spend all my limited time on weeding out the same weeds....

After you weed, then put down several sheets thick of newspaper to keep the area open. Mulch on top of the newspaper to hold it down and look better. This really works.