Hello Namder, that looks a big undertaking and a lot of perennial weeds. I hope you are allowed to cover with mypex or simple black polythene because it would be the easiest and quickest way to do as Bluebell says above, clear the perennials (while growing simple and wide-spaced veg) to give you a clean start by early autumn best year.
This poly cover is only until next September, then compost thin layers and hand weeding.
Any compost/organic matter spread on top before the poly will give richer soil and nicer surface for the future.
