To actually have that returfed is probably not going to cost more than about £50 - £100 in turf.I think the issue really is about sorting out all the sh1t under the grass.It's a common problem particularly on new houses that they have great looking lawns for about a year or 2 then it all starts just dying away as the roots reach the rubble and have used up any nutrients that was in the soil that the turf was laid on.
The reality is hard work of digging out all current grass and earth, putting to one side then removing as much rubble as you can and then put the soil you dug our back mixed with more good top soil and loam and then either reseed of better still turf. I think it might be worth getting soeone to quote you.This time of year landscape and gardening comapnies are always short of work and will readily take on jobs like this much cheaper than they would in the spring and summer peak when everyone want these jobs done.
I know 2 guys with gardening companies who say this time of year is always really slow for them.
HTH