Yep, she doesn't care, but I think you hit the nail on the head with your own idea ~ 'drastically cutting down her wardrobe to just seven outfits since she can't seem to manage her wardrobe. Then she could do laundry once a week or run around naked.' Just explain to her that since she's not ready to accept responsibility you have to take charge and make some changes.
~ Assign a time for her laundry to get done, between "___" and "___" on the day she'll be doing it, giving her a few hours leeway, say on Saturday. Schedule it early in the day in the event she doesn't do it, and tell her she's grounded the rest of the day if she doesn't ~ what 10 year old wants to stay home, not watch TV and miss out on doing fun things on Saturday? Then give her the same time on Sunday, she HAS to do laundry for school... Hide some mismatched outfits for her to "buy" for a couple of $ of her own money when she doesn't take you seriously and neglects doing her laundry, something she'll have to wear but won't like, and schedule something away from home like grocery shopping so she can't just sit in her room wearing it. Scheduling times to do laundry is what they did in my college dorm and an apartment building I lived in, use your time or lose it because someone else has to do theirs. (And do some of yours so her time is really up!)
~ If she wants to exchange some of her outfits for the following week make sure she washes the dirty ones first and then exchanges them ~ just in case she thinks she can outsmart you.
~ As far as whining and complaining, send her to her room to do it, or walk away and go to your room, lock the door, and tell her if she keeps it up she'll lose a privilege like TV, going to a friend's or the movies, etc., it's unacceptable. Treat her like the toddler she's behaving like.
~ Move her laundry hamper to the washer/dryer area, she can place dirty clothing there after her daily shower or changing. The less she has to deal with in her room, the better. And hide the laundry secretly in a trash bag where she can't find it in the event she tries to sneak something out.
~ Make sure there isn't a lot of junk in her closet and wherever she puts clean laundry, my daughter was a packrat :-/ If she appears to be keeping her things better organized consider adding one outfit back a week, and be prepared to trim them back down to 7 if she starts being messy again.
~ I seriously got rid of a lot of clothing when my daughter was starting 6th grade, (11) she had way too much stuff and it was hard for her to keep it organized after looking through everything for something to wear. We kept the things that were most versatile, that could be mixed and matched to give her the most clothing options with the least amount of pieces. She had no problem with it and was still one of the coolest-dressed girls in her class. She also learned how to shop wisely, "what does this go with?"
So exhort your authority, set up some new rules, and, yes, it will pass...middle school or junior high is coming in a year or two!!