Unfortunately, there are only a set number of questions completable for each section. Unfortunately, there is no way to limit the number of attempts on that. What I suggest you do is to either find some other problems online and have them sit down with a pencil and paper, or (if you are not busy, which you probably are) you can try to actually sit down and talk it through with them.

Depending on their ages, it could be beneficial to install browser extensions limiting their online activity, if they are getting distracted by that.

Maybe you could try to make some sort of incentive? I'm not sure what a great idea would be, that will obviously depend on your family, but I'm sure you can think of something.

I hope this helped!

Jermiah, I really don’t get how you don’t see the real problem here; the fact that KA severely punishes you for getting a single question wrong. I really think you should be asking for something where maybe there are more questions, but less quizzes and allowing the user to pass with one question wrong. When people like me got a single question wrong in the past, we were SEVERELY devastated. We would be trapped in a quicksand pit of making one mistake, and going back to fix it. But maybe we make a separate mistake. We fix that, and go back. We attempt it once more, only to get 1/15 quiz questions wrong and have to go back. It takes forever to get it right.
Really, the temporary solution is to talk to your kids and tell them it’s okay not to get 100% mastery. But limiting the attempts is going to be rage-inducing and is going to cause more frustration than it prevents.

If a student does poorly on a quiz, do we want to punish that for life?  Letting them do more study and try again is important so that they will actually do more study and learn.  Otherwise, the student would simply move forward with the learning gap and that gap would present itself later in other courses.

As mentioned, it would be better if Khan had a much larger pool of practice, quiz, and test questions.  For the mathematics courses at least, and perhaps other courses as well, I suggest creating a test engine that can generate and grade test questions on the fly.  This would be helpful for practice questions as well.

Along with larger or automated question pools, Khan should allow students and perhaps teachers to see the entire history of a student's work.  If the student had to take a particular quiz multiple times, the teacher might intervene with some one-on-one assessment time on the topic.  But one of the great outcomes that can come from our technological age, in regard to learning, is to enable students to re-study, and retake exams.  Afterall, the goal is learning; it is not about the assessment.  Assessment, in any of the forms discussed, is only an aid to learning.