I would expect to be weeding a lot for several years and try not to let it get out of hand again!

Pick as much seeds as you can, then pull the weeds, pile them up and burn them when dried. you can do this right on the patch or somewhere else.
This is probably too little too late but its better than doing nothing.

been there done that (:
just make sure next year is a mulching year and you will be fine

I don't know how to use a weed burner thingy, but in my area you DON'T want to at this time of year!
You might cover the area with plastic to kill everything or you can use the round-up that also has the pre-emergent or just a pre-emergent by itself.
I know, I know, don't want to use the chemicals, but sometimes that's all you can do!

Go through the garden with a clippers and a garbage can. Gently cut off the seed heads or pull out by the roots and place them in the can.

Alex

I use large quantities of weeds to make hot compost. Most of the viable seed heads get "burned". I don't sweat pulling an occasional subsequent weed.
You could shallow till the area and plant a smother crop. Around here, buckwheat serves well. Three rotations at around four weeks each will eliminate existing plants, unless there is lack of germinating moisture... the cover crop needs to "jump up" and choke competitors.
All the green manure improves your soil.

I used Preen for the first time this year and I followed the directions and it worked really well. I didnt need to weed at all until August--then it seemed the weeds came out of nowhere and took over parts of the garden yet other beds are still weed-free today. In the places where the weeds grew back, maybe the Preen wore off.

I hope I get the same effect next year--not having to weed during the summer was really a treat for me.