Troublesome weedy grass- please help!

Hi everyone. Long time reader first time poster. We have been in our new build for 1 year and have been slaving away trying to rescue the poorly laid grass. As you can see we now have a seriousssssssss weed issue which we just can’t get control of! (Let’s not even get started in the dead patches but never mind!) We’ve tried weed killers and nothing seems to be touching them- if anything they’re growing with a vengeance! Unusually our direct neighbours don’t have a single weed, but ours are out of control! Can anyone give me any recommendations on how to tame them, short of ripping the whole lot up and starting again?!