Oh, I'm laughing! I was just like that. It's amazing that I lived to grow up.
Here are some thoughts - just off the top of my head (I apologize in advance for how long this is going to be):
Some people, it goes without saying, have more of a bent for cleaning and organizing than others. Some are natural "neat freaks" - and others are natural "messy freaks." I've heard messiness linked to creativity, and I think it's a marvelous thing to consider, but I don't know if it's really true! All I know is that, back in the days when dinosaurs roamed the earth, I would seriously start putting toys away and ended up playing with them instead; I'd start to put books away and end up reading one (well, it was right there); I'd put away art supplies and get an idea I just had to try out with paper and paints. I had paper dolls and I just HAD to put them in their proper families before they could go on the shelf, and you know how that ended. I have always had a perfectionistic tendency, and I've had to get out of that mental attitude.
However... I'm reading over your post for the third time, and I'm thinking there is also a war going on. You're pushing, pushing, pushing the clean-up detail, and she's pushing, pushing, pushing right back at you. Even at my age, which I won't tell you, when somebody nags at me or yells at me I automatically move slower, not faster. Life is too short. Can you call a truce for a bit?
Of course your daughter needs to learn to keep things neat and clean! It's just something everybody has to learn. But it won't get that much worse if you both take a week off from the war; for one week, you don't nag and she doesn't dig her heels in.
During that week, she can think about the fact that she *must* learn to keep things clean, whether she likes it or not. It's just a fact of life. All her friends are learning the same thing at their homes. During that week, you can think about whether she could start learning it in another room first.
Some time during that week (don't tell your girl you're going to do this), go into her room alone, squat down or sit down on the floor until you're roughly her height, pretend you're her, and look around at the room. If it looks overwhelming to you, that's the way it looks to her. Six may be the oldest she has ever been, but it still is pretty young, after all.
Let's imagine that she can start learning to clean up a room by helping you (more or less) wash the dishes and put the food away after dinner. You can talk, casually, about how good it is to put the food away because you always know where to find it again. You can talk about liking clean dishes because eating off dirty ones is blech. (Let her experiment on that idea if she wants to.) No pressure. No lectures. No responsibility on her part beyond being Mama's helper. Just conversation. And while you're conversing, look for the GOOD things she does and praise her for them. Don't overdo it, but say it out loud. Your daughter might listen, because it isn't her room. It's neutral territory, so to speak.
Maybe after a week you'll both be fresher and less upset, and able to talk about what might be done to a room to help make a little girl take care of it. I knew a mother and son years ago who packed up everything of his in boxes and moved it all out of his room for a few days - everything that wasn't an absolute necessity! He knew where it all was, but it wasn't in his room. It was as if those things didn't exist. I remember him saying it was like camping. After a few days, he knew what else he wanted most - a few more clothes that he needed, his favorite toy, and a book he hadn't finished. So he moved those things back in, but that was all. Bit by bit, they added things back to his room until he said, "This is it." They kept the other things in boxes (including his bedroom TV, because he didn't ask for it!) until he decided that he could pass a lot of those things on to someone else. That's a whole lot of work and a whole lot of rethinking, but it worked well in this case. Both of you could try thinking outside the box for other such ideas.
One time you could let your daughter clean YOUR room! She might have some fun with that. Or let her do a part of your room - like rearranging your shoes or a dresser drawer.
Nervy Girl's story suggestion is a really good one! "The Won't-Pick-Up-Toys Cure" is a very cute story and you'll find it in the first MRS. PIGGLE-WIGGLE book by Betty McDonald. (My granddaughters and I love Mrs. P-W.)
The suggestions that the room-cleaning be broken down into bite-sized, detailed segments may be very good ones. Does the bed need to be made? Well, how do you make a bed? Break it down step by step. Ask her to test your steps and see if they're good ones - because bed-making is harder for short arms than for long ones. How do you put books on a bookshelf? What are all the motions?
There's nothing wrong with saying, "After you pick up your clothes, you can ride your bike," or some such thing. Grownups have to do their jobs before they can have their fun, too, so they learn to do those jobs as quickly and as well as they can to get to the other things.
Once your daughter is at a point at which she knows all the motions, would she like working against a timer? Some kids enjoy seeing how much they can accomplish in three minutes.
As you both work together on this, let her know that it's something she will be learning for years, and it's something you've been learning for years, too. You weren't born knowing how to keep a room neat (were you?). You learned - do you have any messy-mama stories? - and she can, too, over time. Again, look for the good things she does.
These are just ideas to think about to help her learn how to do the job without you two having to be at war about it. You won't be doing less work yourself, but you'll be training her in how really to do the tasks, and in another few years she might be able to take much more responsibility herself.