m1469, How are you marketing your piano teacher business? How do you get your students? There are many things teachers do. Website / yelp - some teachers plainly state that they only want to teach teenagers who are truly committed to become a musician to ones who say I love just sharing the joy of music with toddlers. Some flyer in library, grocery store, local schools, many teachers promote themselves, and you need to communicate what students you want.
Referrals with other teachers - some like to teach Jingle Bell to kids who don't practice and some like to teach Bach. Definitely, going back to school helps you find those networks, but also taking a piano lesson from a very established teacher will help you create such networks because s/he knows tons of teachers.
Also you need to grow your business, so you can actually start sending out ones you don't want to teach to others. If you need 20 students to pay bills, try to aim 30 students constantly, and send ones you don't like to other teachers. I know some teachers do local concerts, perform in churches, or even take church music director jobs, to broaden their reach. Also that builds your resumes and richer resumes attract students who are interested in them.
And I agree with nylregy, you should just happily take the money from students who don't meet your standard. You may not be fulfilled, but students are fulfilled with their music interest, that's why they come back. Even they didn't practice, they still come to you, because they want to maintain their ties with music. In their perspectives, they are fulfilled. Let them be. Repeat the same thing you said last time with smile, and if they remember, say "good job", and take the money. And by that, you are actually giving more to the kids. Otherwise, kids would eventually hate music feeling inferior. If they really make you unhappy, really don't take them. Tell them you know a better teacher to teach such repertoires and send them to ones who like them.