As with any perennial weed, your options are physical removal or chemical removal. Physical removal entails either digging the plants up or covering them with a light-blocking tarp to starve them of sunlight. The latter technique might not work very well for any weeds that are mingled with desirable plants, though, since smothering can harm or kill them in the process. The goal of smothering is to deny weeds light for long enough that they run out of root energy stores, though it's hard to predict how long that will take with vigorous or aggressive weeds.

Chemical options center around systemic herbicide use, which is absorbed by the plant (through foliage, in this case) and moved into roots where it kills the whole plant. Even so, more than one application might be needed before treatment is successful, and these kinds of usage directions (like how long to wait between sprays) will be on the product label. Note that these kinds of herbicides tend to be non-selective, so may injure or kill any other plant contacted by the spray, so make applications carefully.

How long it will take to eradicate the weeds is hard to predict, but any regrowth that appears (which would not be unexpected) should be removed (dug up or sprayed) promptly to hasten its die-off.

We do not have medical training, so you would need to discuss any herbicide concerns (or dog eating the weeds concerns) with a vet. The herbicide label instructions should also specify how long after application a person or pet can enter a treated area; if it's not clear, you'd should contact the chemical's manufacturer.

Miri