The 1867 law was universally unpopular. The landlords cut windows into every internal room, but facing the interior hallway-stairwell, so no fresh air or natural light came in through them. They merely robbed the immigrant tenant of privacy. The law was a burden to the landlord and a disservice to the tenant. Nevertheless, it took the city gov't twelve years to improve the law. (Compare the one year that the city took to fix the shadow problem for the real estate industry between 1915 and 1916.)
The 1879 Tenement House Act (now called the Old Law) required windows facing fresh air and natural light. A trade journal, Sanitary Engineer, hosted a competition for the best design. The winner's design conceded the least to the law, the most rentable space to the landlord, and allowed the footprint of the building to extend into any lot without bound. It was the dumbbell tenement, named for its distinctive footprint which allows minimal light and fresh air all along its side abutting the adjacent building. The dumbbell tenement appeared on the scene coincident with the sudden rage for terra cotta, making it easy to identify them. If you have any doubt, you can always look at the footprint from google maps.
The dumbbell had literally fatal flaws. In a fire, the airshaft acted as a flue spreading fire to the upper floors and to the adjacent building. Even worse, the airshaft was dug down to the foundation with no convenient egress. It was not designed for garbage removal. If any garbage accrued, the tenants assumed that garbage was its purpose -- people usually learn from example, and who would educate them otherwise? The result was an unsanitary nightmare. Parts of the immigrant ghetto were virtually quarantined, they were so unhealthy.
Again, it took the city decades to fix. During a progressive age of good gov't -- a moment when the elite Republicans recognized that they couldn't regain political control over a majority labor city without serving labor better than the Democrats -- a New Tenement House Act not only fixed the problem but also changed the face of the streetscape with several unintended consequences. The New Law required that each building have a large courtyard for garbage storage and removal. This yard had to be so large that single-lot landlords couldn't redevelop their buildings without losing space. In effect, the New Law forced the preservation of single-lot buildings throughout the city. The exceptions was the corner lot, which can be built more deeply into the lot and which benefits more from the open interior of the block to allow window access.
The five story buildings were probably all owned by different owners who couldn't redevelop because of the New Law courtyard requirement. If you go to Washington Heights, what you first notice in the neighborhood is the scale of the buildings, much more expansive than the LES. Washington Heights was built after the New Law.
Next up, don't believe what you read.