First of all, mowing, you will be glad to hear, does eventually get rid of it, especially if you do it as often as one is supposed to, if one wants a fine close knit turf.
In the border itself, the only way is to dig out all you can and then wait until the new growth starts. Bruise the foliage and apply the Round-up.
I would like to try covering the soil with black plastic and letting the weeds grow underneath until they are a few inches tall and yellow, then weedkilling to see if as the leaves develop the full chlorophyll it makes any difference to the uptake of the chemical. Where are you John Miller or Hugh Jones for their thoughts on that?