The brown patch could be caused by the conifer aphid, look for sooty mould further down the shoots, this has become a huge problem of the last ten years, if this section of the hedge is dead then it will not grow back, but one method you can use which I have used over the years is to let either side of the affected area to grow out and given time you can wire these in and trim carefully, it works but you have to be patient , you could also dig out a good area at the base and take out the exhausted soil and replace with decent organic compost and pop in another leylandii, but try and find the cause of the problem first, if it is down to pests , thrips/ aphids then provado bug clear is fantastic and used three to four times a year will keep the hedge in good condition.