I would treat this area like a first year planting and mow it to six inches so that the weeds are not able to set seed. Mow it to ground level in the spring. I do not think the new desirable seedlings will tolerate a burn just yet. You'll have to keep after the weeds you listed as many of them are quite prolific.

Thank you Theresa, you make a very good point about avoiding fire damage to the new seedlings. I did surgically remove all of the burdock last weekend, so I at least eliminated that before it went to seed. I am going to mow the area with a string trimmer to ~ 6-12", most of this stuff is 4-6', so I am not sure if the clippings need to be raked off? Should I worry about mowing desirable plants such as yellow coneflower, or just mow the whole thing? Thx...
Chris

Chris, you don't have to worry about harming the desirable plants with mowing. They will recover. But, if you can work your way around the good plants, that would be great for setting seed. The bad stuff will recover as well if they are perennial, so you'll have to keep after them. The advantage of mowing is to prevent weeds from going to seed and to give desirable plants some much needed sunlight as many weeds tend to grow fast, shading out the slower growing natives.

You'll have to remove the cut vegetation if it is smothering your prairie or if it contains undesirable seeds from the invasives in your prairie.