Hi A.,
First, know that you're not a terrible mother for this at all! Being a mother, especially a SAHM who is "mothering" 24/7, is the hardest thing I've ever had to do by a million miles (though, of course, it's the most rewarding at times too). I thought I was the most patient person in the world until my oldest hit the 2½ - 3 yo stage. The whining drives me INSANE too. Every person has their limit and as a mom, especially of a toddler or teenager, it's a line that is constantly being danced on in one way or another. I think I have some sort of sensitivity to noise too, which makes it even worse. (I seem to lose it more with noise issues than any other kind, other than one of my kids injuring the other.)
Although it's a maddening stage, know that it is completely normal, and in a sense, a healthy stage. She is still trying to figure out the world around her and how to make it work for her. Of course, this also means that you need to teach her that whining is NOT the right answer, and help guide her to a better way of communicating her wants and needs.
I think what you're doing now is a good first step. It's crazy hard now, and I know it feels like it will never end, but if you stick with it, it will show results. I use kind of a two to three step program with my boys.
Step #1: Similar or in addition to what you're saying now, get down to her eye level, ask her to look at "mommy's eye's" and calmly say, "If you need something or want something, you need to use your big girl voice". The eye-to-eye conversation really helps them focus on what you're saying and it may also calm her down for a moment to listen. Then give an example of the RIGHT way to ask (when it is clear what they want), so you're showing her the right way instead of just ignoring the wrong way. Once she asks the right way, respond positively and quickly. Even if the answer is no, giving a positive response will make an impact. If you know the answer's going to be no in advance, you may want to preface it by saying "you may not get this still, but I will respond to you and we can talk about it." Or something along those lines.
Step #2: If the whining continues, explain that she will go in time-out (I like the whiny corner idea below too). Explain that she can go sit in her room (or wherever) until she wants to come out and talk to you in her big girl voice. (When she gets older, you may want to omit this step as she shouldn't need warnings anymore, but at this stage, I think a warning is helpful and fair for her mindset.)
Step #3: If she's still standing there whining, pick her up and put her in time out.
Personally, I don't think ignoring it is the best answer, although I completely understand getting to that point and admittedly have gotten there myself before. If you do the hard work right off the bat and get a pattern of communication, then positive reactions or consequences set, then you should be able to significantly cut back on the issue. (It will likely not completely go away for a few more years though.) Ignoring is only teaching your daughter to ignore her own problems (as you are her biggest example) instead of dealing with them, and is not addressing the problem at all. Just try keep in mind (somewhere in the little sanity you have left in these moments) that there's a much bigger picture at hand. As absolutely maddening as you feel sometimes (and as legitimate as it is), try to keep in mind that your daughter is learning how to deal with the things in her future directly from you. If you were to imagine a conversation with her when she gets a little older to try to help her handle something in her life that she doesn't like - what advice would you want to give her? Or think about what advice you would give a friend on how to deal with a problem they're having. What do you think is ideal? That's what you need to start teaching her by your own example now. This is the most critical learning stage for things like this.
Ok, I've babbled enough now.
I wish you all the best in whatever you decide to do! Good luck and know that it will get better!!!