Motivating the Unmotivated

A challenge that we all face is motivating the unmotivated learner.  It seems that high school education culture has grades and college as a high motivating factor for most students.  This is a problem as high school and college no longer seems to be about educating and learning for the sake of learning and to be informed about a body of general ideas, but rather a training ground where students are motivated to "make money."  This bothers me, but is not the point of the post.
How to motivate the unmotivated...
I want to devote my thoughts to the unmotivated or underachieving students as the focus for this post.  I teach the regular-level classes (non-honors) and have a range of students from those who really enjoy math and routinely master the material, to those who loathe coming to class.  One of the first things that I do in class is make intentional relationships with students.  I try and get to know them and see them for more than who they are in my class.  I am thinking about trying to reach the lowest 10% of kids here; those who dislike school; those who hate math.  Often, I can develop a relationship such that when I request a student pay extra attention to homework, they will comply.  I can request that a student come in for additional help, and usually they will.  What about the kids who won't?
Here is my struggle: motivating the students who hate math and do poorly.  This past year I had students in study hall that I could help who were lower achieving.  This took daily work of my looking over their shoulder to ensure that homework was complete and correct before they even came to class.  I also used this time as remediation with them so that after poor performance on assessments, we could review and remediate weak areas.  Then they could reassess.
Two specific examples of student interactions this year.  Student A struggled with a D all year and with some increased work in the trigonometry unit was able to pull his grade close to a C.  As final exam time came close, I made a contract with the student.  I highlighted 40% of the 10 page review guide.  The students had to complete all highlighted portions of the review with correct work shown for each part of the problem before I would let him take the final.  If on the final exam day, he came to school without the highlighted portions complete, I would make him work on those questions while his peers took the exam.  When he finished he was eligible to take the exam during the make up exam period.  This strategy worked for student A.  He came to the period with excellent work on the review packet and was able to take the exam and keep a D in the class for the semester.
Student B dislikes school more than student A.  During the trigonometry unit I gave a 20 point formative on finding exact values, then a 20 point formative on solving for angles.  (I think of this as working forward then working backwards.)   After the two twenty points, we put them together for an assessment working on both skills worth 40 points.  Any student who earned less than 15/20 was told to come in to remediate and retest on either/both of the first parts.  This would ensure a basic understanding of the material as every trig assessment for the next two years builds on these concepts.  Almost all students eligible came in to work toward mastery, such that the median score for the combined assessment was 37/40.  Student B did not take advantage of this opportunity.  So, instead of taking the 40 point quiz, I had him remediate and retest his previous scores of 3/20 and 11/20.  After he retook these, then I had him take the combined assessment, earning 32/40.  What a success.  Unfortunately, student B still earned a D in the class such that the final exam was the determining factor of whether or not he passed the class.  He did not complete the review, nor did he come to review sessions, thus earning a 35% on the final and failing the class.
Why didn't I make the contract with student B too?  At what point does the responsibility for wanting to learn and succeed take over?  These are juniors in high school.
Some colleagues have instituted a two strikes homework policy.  Students who miss two homeworks must come in during lunch or after school to complete the next day's work.
Some colleagues have adopted A, B, not yet.  This is interesting, but oh the retesting and remediation.  With 130 students I think that I would be exhausted and not physically be able to remediate all of these students and make enough new assessments.
I wonder if you used standards based grading.  Would this be an improvement?
What about if students completed units of study as projects of the body of knowledge.
Education is changing so fast.  I struggle with having students learn math because it is important and the amount of real application involved.  Truthfully, there are some lessons where the application seems so much harder, I don't know if there is even benefit to say "with 10 more levels of difficulty you could use ___ in the real world."
I would love some thoughts and feedback here.