This building survived the fire in 1776, but it wasn't being used as a school at the time. It had been shut down under the orders of the Continental army and turned into a hospital. It didn't open again until war ended, when it was given a new name - "Columbia College". Another example of the word "King" falling out of favour among Americans - unless it involves burgers, of course, which just sums you people up.
The college stayed here until the mid-19th century when it moved to Midtown. The building's cornerstone was moved to the new location, but the rest was torn down and the land underneath it was sold at a tidy profit - which is the history of much of downtown New York.