Posted: 05/03/2015 at 20:26
We had a lawn of almost exactly the dimensions of yours, with similar problems. I wasn't up to doing it myself so got somebody in to do it. They did rotovate the whole area - 3 times. Up and down, across and then up and down again to break the soil and grass up really well. Thoroughly levelled and firmed and then turfed. That was done in June, far from an ideal time but we kept it well watered and everything is good now.
Turfing is obviously dearer than seeding but the practicalities of getting round the garden if it was all seeded made it a no-brainer for us.