I'm not much of a gardener, but many years ago we had an allotment infested with couch grass. I'll start by saying if you want to get rid of it, you'll get very fit.
The method of covering it you describe in the OP will work eventually, but you're talking many years.
Instead of weedkiller, remove the top matt of grass manually with a sharp spade, taking care not to remove too much topsoil. Then work carefully with a fork to loosen the soil, not too deep, then you'll be able to grab the heads of the roots and gently pull them out. If the soil's not too heavy and you've loosened it in the surrounding area, you'll be able to tease out the long white roots. Beware that every single tiny piece of white root left in the soil will regrow, and quickly. Don't put them in the compost - either dry them for a few weeks on a wire net in the sun, or put them in the bin.
Leave for a few weeks then come back and do the same again.
At that point, you have a choice of putting down the cardboard (in the past old carpet was also an option, but nowadays it will be synthetic and/or have a foam backing) and grow your wildflowers on top. Alternatively, sow the wildflowers in the nicely dug-over soil and let the couch grass that's left be a part of the wild meadow.
I guess now is not the time to sow anything, so you could prepare as above, cover with whatever you have to hand (black plastic works well), then in the spring when you lift it off you'll have a matt of white roots trying to find the light that can be lifted out fairly easily, then continue as above.
