It's a waste of money using herbicides when the weeds aren't growing. Don't rely on Roundup too far; it'll kill the more delicate weeds like annual grasses, but the tougher ones will come back. At that point, they'll have no competition, and the last state of the plot can easily be worse than the first.
There are three practical ways to tackle it. Dig the weeds out and compost them, which is hard grind, but relatively quick. Black plastic for a year. Dig it in round the edges so the roots don't just run under it. Or mow it for a year, which will kill most things. A combination may be best. Dig what you can, and black plastic or mow what you can't. Keep digging it over as you can.
Never throw out green material. Compost it, and put it back on the garden.