If you want to go down the chemical route then use a Glyphosate spray, such as Roundup. However the soil will be full of weed seeds that will immediately start to sprout once you remove the competition from the existing weeds - remember the old saying: "One year seed, Seven years Weed!"

If you prefer to be Organic and can wait a little longer then simply smother the whole area with thick layers of old carpet (readily available from your nearest recycling plant!) or black plastic - which will "cook" and kill many seeds as well as the weeds.

I think you only real chance of clearing this, is to dig it over and get the roots out covering in carpeting/plastic suppresses but does not kill the roots.

thanks