The photo and description of the weed sounds like Canada Thistle, a very common and hard-to-control invasive species. The most effective approach is probably going to be spot-treating individual thistle plants with systemic herbicide, which the leaves absorb and move down into the roots. That transport in the sap likely is most efficient around the time the plant comes into flower, but you can try any time as this will probably require multiple applications to achieve complete eradication. Follow label directions on the herbicide of choice (make sure Canada Thistle is included in its list of weeds controlled) with regards to dosage and the timing of repeat applications. Dieback from any treatment might take time (perhaps 1-2 weeks) to manifest since it takes a while for the chemical to move into roots and cause damage, and don't remove what look like dying thistles until they have had plenty of time to succumb.

Miri