"Quite" is an adverb that modifies "the way I had expected." It suggests that the assignment was completed in a manner close to, but not fully aligned with, the expectation. The complete sentence is: 

"The student did the assignment, but not quite the way I had expected."

The word "as" can precede "expected" in sentences like "He came as expected." In such cases, "as" serves as a conjunction, linking the action to the expected outcome. However, the phrase "the way I had expected" already specifies the manner of the expectation. Including "as" creates redundancy and leads to an awkward construction. "But not as the way I had expected" becomes unnecessarily clunky because "as" and "the way" are overlapping in function.