The only way we have found to get rid of any unwanted plants including grass and dead nettle has been to cut off as short as possible, cover it heavily with mulch, and sometimes a tarp over the mulch to smother it out. This can eliminate both the root and the seed. Even then we will need to continue on areas that did not get killed off completly. Tilling can bring more seed up to the surface. We cover an area that we intend to plant over with a thick layer of shredded leaves and after the leaves decrease in volume we put top soil over the mulch and plant in the clean top soil. Sometimes we put a wire cage over the area and fill it with the chopped leaves. Then remove the cage after the leaves settle. The objective is to eliminate the light that is needed for growth. It is also easier to remove stray weeds from the treated area because the soil is looser. Can you pull it out from around your plants before it reseeds? It is a common ground cover weed. We have used round up on poison ivy and other woody plants but have not used herbicide on the ground cover weeds. We will sometimes begin with one small area and expand it little by little around the borders. This has worked well for us. EP

This terrace had been covered with grass, dead nettle , and black eyed susan. We treated it as described above. We needed to remove the mulch from around the shrubs immediately in the spring to prevent smothering the shrubs. Hope this helps. It takes time and patience. EP

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This is a tool I have found to be very beneficial for cleaning up weeds such as dead nettle, henbit, and chick weed. It is also known as a scuffle hoe. These weeds persist as ground cover in winter so it is important to try to eliminate as many as possible during winter months. We hope you will share more about your challenges and successes with Ozark gardening. EP

Here is a link that might be useful: stirrup hoe

Have you considered a pre-emergent chemical treatment that prevents seed germination? There are selective ones that prevent only broad leaf weed seed germination. and do not affect existing plants. The link below list some commercial types available and their active ingredients. Always follow label directions for optimum results. Incorporating the chemical in the top 3 inches of the soil and then applying mulch has proven to be very effective in controlling even the most difficult weeds. Also, have seen the use of several layers of newspaper, covered with a thick layer of mulch to be effective in controlling weeds in raised beds.
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Here is a link that might be useful: Weed Wizard