how to kill couch grass

There's not really a lot you can do with it at this time of year, spraying will be ineffectual as the grass is practically dormant. You need to wait until spring when it is actively growing.

I've not got so many years experience but much of what I have got has involved battle with couch grass and creeping buttercup.
From what I've seen the best method to get on top of couch is to dig by hand and remove the roots as you come across them. As GG says, rotavating will just chop the roots up into lots of little pieces that will all tend to become viable couch themselves.
Covering with black membrane will weaken the couch - I found on my soil it tended to make it easier to pull large amount of root in one go (once the membrane was removed). I never managed to completely kill it off using membrane alone - I guess the root system was spread out past the membrane...
Weed killer seems to have less impact on couch than you might hope for - and at this time of year really wouldn't be effective.
Sounds like the area of land you are talking about might be a little too big to work by hand... its a bit of a swine but you should be able to weaken and eventually kill with (a lot of) perseverance. 

Pip pip,
Balders

While my garden has been frozen solid or covered in snow I have resorted to gardening on YouTube ,I watched a video by a chap who discovered by accident that covering the ground with Barley straw killed the grass, he did some research and concluded that as the straw broke down it released a chemical that killed grass roots but nothing else.
I don't know if it works and have no need to try it but it must be Barley straw

Sorry I made a mistake it's not on you tube it's an article in Marches Kitchen garden magazine  .

I have read somewhere that thickly sown turnips will stop couch grass spreading. I don't think it kills it just suppresses it.