I have two general rules about chores and relationships. One is that if you can possibly solve the problem by throwing money at it (buying a dishwasher, hiring someone to clean), that is money well spent. Two is that you are not allowed to complain about how someone else does the chore -- either do it yourself or shut up. Those are your only choices. If you don't like the way she washes dishes, do it yourself. If it's not important enough to you for you to do it yourself, it's definitely not important enough to argue about. And probably she hates the way you vacuum or something, and she should do that herself. And you'll probably end up each doing half the chores once you figure out who cares about what.
The situation is somewhat complicated if the other does something that makes your chore harder, like how she doesn't rinse things she puts in the sink. We have the same problem at my house. While I can sort of see the point, I really can't get all excited about rinsing something so perfectly that it practically doesn't need to be washed anymore. So I make an effort but it's apparently never good enough.
I think you have to meet halfway - she needs to promise to try to remember to rinse things at least half-assedly and you have to promise to shut up about it not being absolutely perfect. And you need to believe in each other's good intentions - as long as she's making an effort you have to believe that she is doing what you asked, as far as she can do it.
But if you really need the dish-rinsing to be perfect, then you go back to rule 2 above: YOU need to rinse her dishes after she puts them in the sink. Because those are your only choices: Doing it yourself or shutting up. Or, of course, living alone.
Sometimes the application of Rule 2 is amusing. The Technical Staff actually writhes in misery when I load the dishwasher on the occasions that he doesn't want to do it. It's SO hard for him not to say anything about my dishwasher-loading incompetence. In my case the difficulty is usually the 'shut up and eat what's in front of you' application of the rule, when I haven't done the cooking.