Try smothering them. First mow them down as low as you can, then cover with a layer or two of cardboard, then a few inches of fine mulch. In a couple of months, the area should be plantable and you will have avoided weeding. There could still be live seeds below all of that --just avoid tilling the ground and just dig where you want to plant.

Once you've killed them, you'll need to take measures to prevent. Overhead watering of planting beds is the biggest problem for weed growth. People also water planting beds just as frequently as they water lawn-- this is wrong. Planting beds only need an hour of drip irrigation, once per week. The best thing you can do to control weeds is to control water and switch planting beds to drip.

Also, you'll need to top dress the beds to maintain control and prevent weed seed from getting the sunlight it needs to germinate. DON'T put down weed barrier fabric-- instead use 2-4 inches of compost, mulch or I guess rock if you like that. These two things cut weed growth by 80%. I have personally tested this in my yard an it's amazing how well it works.

To get real control of a big problem, you do have to use chemicals. If you want to be very targeted in the application, use the new Roundup gel stick so you can selectively kill (the mallow and white top especially) down to the root. Once you have control of the space, hand weeding (along with the 2 measures above) can be an effective to maintain control but it's hard to hand weed enough to get control of a space.