The weed pictured here is Canada thistle (Cirsium arvense).

Canada thistle is a creeping perennial, reproducing largely by rhizomes (i.e. underground stems). Canada thistle can also produce seed, though this is not the main means of reproduction and spread. Canada thistle has lobed leaves with smooth upper surfaces and sharp-tipped spines on the margins. The roots of Canada thistle have been measured down to 6 ft, with the majority found between 8-16 inches. 

Removing the aboveground growth can eventually starve the root/rhizome system, but this often takes several years (multiple efforts per year) depending on the level of establishment.
Impermeable mulch (plastic mulch) may help, but if there are any holes in the mulch for desirable vegetation, Canada thistle will likely exploit those. Organic mulches (e.g. wood chips, bark, etc. ) are not effective.
Herbicides are often the fastest option to gain control over a Canada thistle population. There are some measures you can take to best protect the desirable nearby vegetation. The one I have used most recently is to cut the plants at the base and paint concentrated herbicide on the cut surface within a few minutes. In your case, concentrated glyphosate is likely the best option. You can mix this herbicide as directed as well for a foliar application, however anytime you spray you risk off-target movement to your desirable plants, so I do not recommend that in an ornamental bed.
When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid green plant material, exposed roots, and injured bark of desired plants. Third, glyphosate is relatively safe in the environment when used as labeled. It adsorbs strongly to soil in most cases (i.e. clay and organic matter), allowing even sensitive crops to be planted shortly after application; meaning no carry over issues are expected. Fourth, glyphosate alone can take up to 14 days to show full activity under ideal growing conditions. Retreatment of the area may be needed depending on the degree of infestation. Glyphosate is most effective for perennial control in the fall but can be applied anytime the plants are actively growing (temperatures consistently above 50F). Finally, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).
Glyphosate products are being phased out so you might not be able to find these in the box stores.  Roundup products are available but you have to read the label to see what the new active ingredients are.  Some of the new Roundup products available for broadleaf control have limitations on planting new plants after application.  It is important to read the label and follow the directions since these products have changed.  

Thank you so much for your answer.  It looks like I will have a hard time finding this!