If an area's died back it's unlikely that it'll send out new growth, you need someone experienced in conifers really, someone like Bluespruce, PM him perhaps ?

Can't answer because you haven't said what's wrong with them - if they're completely brown all over and showing no growth, they're dead. If they have brown patches, as Louise says, those areas will not regenerate, although the plant may continue to grow in other, green areas.

They might be suffering from the disease that's affecting lots of leylandii at present. It might be time to start thinking about what kind of new hedge you'd like instead, as these brown patches will not green up again.

Agree with Steragram's comments - shame, its a nicely shaped hedge.