I don't think this is a broadfork issue.
How did the cardboard and mulch make things worse? And was it because of seeds in the mulch?
What are you seeding with to out-compete the Canada Thistle and Joe Pye weed? I don't know much about the latter, but in the case of the former, I know that many people prefer to just pull those up as they are found.
What are you using for green manures, and of that, what comes up in the shortest amount of time? I would take that and make sure that it's well-represented anywhere you're looking to extirpate of the aforementioned two. Buckwheat is a go-to for many, but it depends on your specific conditions.
There are many organic control measures listed on Wikipedia under
Cirsium arvense regarding the thistle. As to the Joe Pye weed, it's nowhere near as resilient as the thistle. I would just plant heavily in green manures, grazing or mowing before the thistle gets to seed. Rinse and repeat. Doing this two or more times a season will weaken the organisms, allowing everything else to finally outcompete them.
Plus, tonnes of chop-and-drop. Mow, maybe sprinkle with any needed minerals, and fork it. Fork it good and long and hard. If you improve your soil away from the conditions in which thistle thrives, it won't come back.
Now I realise this is better advice for a pasture, but it still applies in areas where your weed pressure makes
gardening impractical. If you've planted a garden and you get thistles and Joe Pye weed coming up, the best you can do is chop it out just before it blooms (not before; you want them to put its energies into blooming, so less is left for recovery), and keep an eye on it, chopping it out every time. Eventually, if you get the timing right, it simply won't have the
energy left in the
root zone to regrow.
-CK
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
-Robert A. Heinlein