My DD went through a whining horrible phase at about the age of 8 and frankly I felt like I was right on the edge of actually physically slapping her some times. I've lost count of the number of times I have snapped and shouted something like "FFS, go to your F-ing bedroom, NOW". Obviously this is utterly unacceptable and I'm not proud of it, but they do truly drive you mad. It wasn't even what she said, it was the way she did it, it was perfectly designed to get on my last nerve.
Then my partner (who is not her Dad) decided to get involved, which he usually wouldn't. It was such a total relief.
He sat her and I down at the table and explained to her perfectly calmly that whining does not work. It only elicits a very negative response, if it elicits a response at all. He told her he understood that she did it because she didn't feel like anyone took any notice until she got upset and being upset was not nice for anyone, so he wanted to think of a better idea. He explained to her that whining would, from now on, be ignored (along with mubling, which she also did to annoy me) but that polite requests would be considered and wherever possible granted. Then he explained to me, in front of her that she whines because she thinks that being polite doesn't work, so I have to do thinks differently and make sure being polite does work too, in order to hold up my side of the new rules.
At first when she asked me politely for things I didn't really pay attention to all of them, or said no out of habit, which he pulled me up on by saying "she asked politely, be fair". I would then appologise to DD and say "yes you did, I am sorry" and then I would reconsider, giving her my full attention when she was speaking. Reconsidering did not mean she always got her own way, but when she could not I would at least give her a good, logical reason why not. Any whining was dealt with immediately. OH would say "you are whining, we aren't going to listen until you are polite" and she would often just stop, take a deep breath and ask again. When she asked sarcastically she was told "that was not polite" and not answered. When she asked nicely she was answered nicely with our full attention. If she carried on whining the only response she got at all is "you are whining, go to the naughty step and stay there until you can be polite again".
At first we exadgerated the niceness and granted every possible request. So it was still no to chocolate before dinner time but it would be yes to getting up from the PC to turn the TV up. Now I would probably say "bring Mummy the remote and I'll turn it up".
I think you have probably got into the same hole I was in, when everything DD says feels negative and you feel like you just want her to stop bothering you at all. It isn't though. A lot of the time she will be using wailling as she feels polite didn't work, which is perhaps what happened with the TV. She needs to know that as of now only polite will work but you also need to go that extra mile to honor the new rules too. If you can get your OH onside as the referee so much the better while you get started.