my son is 3 now, and what has worked really well with him is to - ad nauseum, every single time if you can - just follow up her whine-statement with the words you'd like her to use in the voice you'd like her to use. if she doesnt just repeat it like that, you can tell her you will help her with X if she can ask you in a normal voice. i think it helps them to name this behavior (ie: your normal voice / your whiney voice). a huge part of this understanding for my son was when we told him we actually cant understand what he's asking for / telling us about if he's whining.
one thing that also helped tremendously with my son was helping him to take a deep, deep healing / cleansing breath when he gets tightly wound about something. we always do it with him. he can now stop himself from having a big meltdown often (not always). its like having a tool box he can take out this tool from and use it to calm down.
so, depending on the moment, you can say things like "I'm right here to listen to you when you're ready to ask me in your normal voice", "lets take a deep breath and start again so I can understand what you need to tell me" or "You know that I can't understand you when you use a whiney voice, so I am going back to washing the dishes, and when you're ready to tell me something in your normal voice I will be right here to listen".
Most of all, remember that if you are CLEAR and CONSISTENT in your not responding to her whining (while explaining why), she will go through this phase and come out the other side not being a whiney kid!
Good Luck, Mama! My best friend is in a very very similar life situation as yours with her 2 year old daughter, living with mom, single working 2 jobs, ex just came back around... if you would like her email let me know, I know she would love to connect with someone to share the struggle. Stay up!