We are all in the same boat, the HMS Whine. Here are some stategies that I have found have worked (a bit).
When dd asks for something in a whiney voice (makes me want to scream and CERTAINLY not give her what she is asking for), I say "ask me again in a happy voice" This I had to model for her for about 6 months solid almost every time she asked for something.
I would say "Please Mummy, can I have a drink/ a turn / a cuddle?" in an absurdly elated tone and only gave her what she wanted when she repeated it in a way that was vaguely non whiney. She is now so well trained that if I just say 'happy voice?' she does it. However, it is getting the first request to spontaneously issue from her mouth in a polite and non whiney way - that is the challenge. It will no doubt come one day, perhaps when she is 20. Along with please and thank you.
If it is something that I do not want to/ can't give her, I say no, plus the reason why not. And I mean it. I saw a Panorama documentary when she was a baby called 'how children get what they want' or something like that. What I remember most clearly is a mother going round a supermarket with a 3 year old who whined for everything - crisps, sweets, toys, biscuits and Mum said no no no all the way round. Then right at the end, she gave in and got her something. The shrink pundit on the programme said, and this has always stuck with me, that the child whined because she knew that eventually it would pay off. Rather like putting money into a one armed bandit, you keep doing it because you know that maybe, maybe, if you are lucky, you will win the jackpot. Whereas if you knew that all one armed bandits did was show a different set of fruits at each pull, you really wouldn't bother with them.
So, what I have always done is say no when I mean no. If she cries I tell her that that's fine, she can cry because she's upset or cross, but that it won't make any difference. Such a cow, I am. But it works - once she is told no, although she sometimes cries, she never whines on and on after I've told her.
Someone once told me 'you can't ignore whining, but you can pretend to'. Easier said than done but it is a good mantra to repeat to yourselves.