Those are common weeds that need to be removed before planting. But...since it wasn't done before hand, and you don't want to start all over...the only options are removal by hand or chemical means. Hand pulling is better, but takes longer. You will have to be very careful if you use something like Weed B Gone, making sure you don't touch any of the plants.

Weeding is a constant in perennial borders. Just do it on a regular basis...even 5 min. a day is fine...and you'll stay ahead of the game. For now, there aren't any easy answers. Start pulling. :)

Since it's only been a couple of weeks, I'd bet you can get rid of those with vinegar.

I always get the 9% vinegar, available wherever canning & pickling supplies are sold.

Here, it's about $1.83 a gallon.

Spray it full strength on the weeds on a hot sunny, preferably windless, day, let it heat up, & rake the weeds out the next morning.

The two weeds I see are purslane (Portulaca oleracea) and a grass which I think is Panicum capillare. Both are shallow-rooted annuals that probably did not come up from the ground below. The were either in the new dirt or the seed blew in. You can remove them with a hoe very easily. Purslane will re-root readily if left on the soil, so it needs to be removed. Vinegar would work well also.

I think that your best bet is to just hand pull them, making sure that you get all of the roots if at all possible. You could spray them, but it is better to hand pull. Like Anita said above, just 5 min a day is fine.

I like to put a few layers of newspaper on top of the soil, then put mulch on top of the newspaper. It really stops the weeds. You could pull the mulch aside and try this in a small area to see how it works for you.

Yep, the newspaper trick really works wonders. You just need to leave enough room around your desireable plants to give them room to grow. I use about six layers and wet each layer to keep them from blowing away.