i read your "what happened" section and im discouraged that you received so many negative posts :( :(
i wholeheartedly support you raising your kids the way you do. even kids need to know that they have to be responsible for chores around the house and etc, because you will NOT be following them to their dorms and doing their chores there will you? you dont exactly want to be footing the bill for a single room because your child cant keep up their end of the bargain in a shared room situation :P - everyone has responsibilities, everyone has to be better at keeping track of their own time and responsibilities, and you are spot on for wanting that for your child.
your daughter sounds a lot like by brother (11 years younger) - hes 16.. and he takes forever to do something. we had him take the flag page (www.flagpagetest.com) and it turns out a part of his personality is to find the best, most efficient ways to do something, and if he cant think of it, it prevents him from doing it. it sounds like an excuse for being lazy, but i think his mind is really working on a good solution (even though that sounds crazy - i mean is there really a more efficient way to do the dishes?) LOL.
what you could do is look up things by dave ramsey. this might help you because he does encourage "enlisting" kids to be paid for certain chores. absolutly; they should do things because they are a part of the family, and they absolutly should have responsibilities that they do not for money, but because it is their responsibility to do it. but there are many things that they do because you ask them to; it would be adventageous to pay them for; especially the older ones. it helps them to learn that money must be EARNED, it doesnt come from just being handed over when they think they need it. dave ramsey (if i remember right) gave his kids 1$ or something, they had a payday once a week, and they did the right things with their money; they gave to church/charity, put some in savings, and the rest was for spending.
not only might this help you to get them to do certain things, but its a financial lesson too, and looking at the world and monitary choices people make (going into debt for things that we could easily and more sensibly save up for) this is something every child should be learning one way or the other (and dont rely on school to teach it!)
anyway, good luck with this. it is hard to figure this out. but you must remain calm, see if you can talk to your daughter, under a relaxed circumstance, like when you are driving or something. figure out what she needs as well, how she feels, what shes thinking while doing the dishes, or whatever. see if theres something she needs help with. when she loads the dishwasher, do you (or someone else) go behind her and redo it? does she do it "wrong" ? this could be causing frustration and making her feel as if no matter how she does it (or how long it takes) its still not "right". LOL. im not saying you do this, but try to think of ways maybe she might feel slightly unappreciated for doing it.
lastly, dont nag. mark gungor talks about ways to get a man to do what you want. there are several ways to do this, but one of the biggest things is asking the right way. dont ask them by belitting, dont ask them by being angry or nagging, dont ask them by constantly reminding. perhaps you could give a time limit for a child to do something, if they dont do it, perhaps offer it to another sibling and give them a "bonus" for doing the chore. this might just mean that said child gives up and figures someone will do it for her if she doesnt, and no big deal, but we all know it has to be done by someone, and it might work? i dont know. :(
anyway, im sorry this is so tough. being a mom is sometimes so frustrating that we dont know what to do. try having a date with your daughter, like i said, and see if you can figure out whats eating her :P