Apply a systemic herbicide, such as 2,4-D. These non-selective weed-killers kill any vegetation they come into contact with. Apply by dabbing the herbicide on several leaves or spray the plants at close range on a calm day. The plants will begin to wilt in 24 to 48 hours and will be completely dead in about 10 days. Occasionally, the plant will regrow because the chemical did not reach the entire root system, so a second application may be necessary. Follow the manufacturer's directions and recommendations.
Mulch mint or lemon balm with a material that completely blocks the sun's rays. First cut the plant back to ground level. Cover the stubs with layers of cardboard or newspaper, a piece of wood, old carpeting, or black plastic. This will prevent the plant from deriving any nourishment from the sun and keep it from growing. For the best chance that your mint or lemon balm will die completely and will not regrow, leave the mulch in place for one growing season, plus one dormant winter season.
Dig out the offending plants with a garden spade. Drive the spade into the soil in a wide ring around the base of the plant. This will ensure that you also remove any underground runners or roots that are venturing out from the base of the root ball. Dig down about 12 to 18 inches to ensure you get all the roots.
Monitor the spot for the next few years. Remove any new shoots that emerge from the ground, most likely the result of small pieces of roots that were inadvertently left behind when the plant was dug out. Applying a deep layer of mulch after digging out mint or lemon balm will help discourage any shoots that make it to the surface from growing or thriving. Use a 2- to 4-inch layer of hay, straw, shredded bark or wood chips.