First, I wanted to clarify that ther are exceptions when I say it is usually one or the other (chol or kodesh, but not both). For some reason, the girls schools that I know about more closely approach the ideal than the boys schools. Currently in the NY area I am aware of 3 girls schools that excel in both. TAG in Far Rockaway, Manhattan HS, and Bruriah in Elizabeth. I am most familiar with TAG, but I have heard very good things about the others.
Nevertheless while the above do prepare well for the state requirements, they are not what I would ideally desire which is allowing kids to actually do their own novel research. However, that requires advanced lab facilities and instructors which is very expensive. Also you can look at Stuyvesant High online course catalog and see what is possible with the right resources.
However, the boys schools are a different story.
Where I went to school is today considered modern, but back then, it had gedolei hador teaching kodesh (no exaggeration, they wrote more about 10 sefarim apiece on shas). It also had a rock solid secular pgm, and kids got into Ivy League schools regularly. They had rigorous chem and physics lab components in their courses. It did not have advanced research, but otherwise was excellent all around. The only drawback is that it is and was coed (although the limudei kodesh was completely separate.)