I think a combination of hard work and then selective weed killing is the way to go. Dig it all over taking out as many weeds by their roots as possible and then wait a few weeks for the deep rooted ones to pop up new growth which you then spray carefully with weed killer. It's not that big an area - if you can set aside an afternoon you will be surprised how much you can get done - 18ft x 9 ft you should easily be able to dig more than half of that if not all of it, in an afternoon. Think of it as saving on your gym membership.
'Optimism is the faith that leads to achievement' - Helen Keller
I think we are reaching a consensus here, weaken it by digging out what you can. Use weedkiller carefully on the regrowth, paint it on if you do not want to spray. There are alternatives to Glyphosate if you are worried, but as has been said used properly it is a tool that works just follow the precautions stated and all will be fine.
Thanks for all the replies everyone. I think, based on the location of some of the weeds (being by walls that I can't get under/around), and one being deeper than my shoulder and showing no signs of stopping there, I will probably get the old weed killer out and then keep at and then also keep on top of any other new weeds with a hoe (that I need to purchase)
Spray then leave for 2 weeks, use the back of a rake to knock the dead weeds down and spray any new growth. 2 weeks later knock the dead weeds down and rake over the whole area. Cover the whole lot area with cardboard and plant through that.
next year will be much easier as you have deprived the weeds/seeds of light.
I would use glyphosate and digging. But if you want a raised bed then dig out what you can then build a wooden frame, put a layer of thick cardboard in the bottom then a layer of rotted manure then fill with compost - then plant!
Dordogne and Norfolk. Clay in Dordogne, sandy in Norfolk.
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Spray then leave for 2 weeks, use the back of a rake to knock the dead weeds down and spray any new growth. 2 weeks later knock the dead weeds down and rake over the whole area. Cover the whole lot area with cardboard and plant through that.
next year will be much easier as you have deprived the weeds/seeds of light.