There is no easy way to deal with this at this stage. I don't use chemicals on my patch unless there is a compelling reason.
This in my opinion is that compelling reason.
Cut them down now and bag them in black bags for disposal. Allow the plants to start growing again, wait for a dry day and spray them with glysophate - it will take at least two weeks for the new leaves to yellow and will kill the plants that are there. There will be new thistles in the spring - this cannot be avoided, as many seeds will already have taken hold. That will be the time to keep on religiously digging them out. Always be on the watch and persist with the constant weeding. Never allow them to make new seed heads.
There will be no instant cure, so accept that.
Your ground will be safe to use next year as long as you keep on working it. You can then use it for vegetables or anything else.
A rototiller is not a good idea. You need to use a digging fork and when weeding finger and thumb work best - the joy of gardening....
If I were you I would put some energy into this now - the longer you leave it, the worse it will get.
The patch is not that big. When I started on a similar problem a few years ago I managed to tame 4 acres!
I'm afraid that one years weeds means seven years weeds - you'd better get cracking
By the way.... I nearly forgot, a big welcome to the forum
@Mamarichey - keep us updated won't you, I for one will be looking forward to seeing the transformation
