An instrumental reason is that form of reason humans have adopted to to deal with a threatening nature. It resulted in the progressive mastery of nature in a disenchanted world.

Adorno's microanalysis of a deformed ethical life of private existence points to a disenchanted rationalized morality in a disenchanted natural world that is represented as a casual order knowable in mathematical terms.

It is this rationalized instrumental reason that is deployed in modernity to rationalize the major institutions of society into the modern bureaucratic legal-rational form---Weber's iron cage of modernity dominating persons. The cage is iron because the main forces of modern life, science, capitalism and bureaucratic organization are triumphs of rationality and so the mind has no purchase to attack them.

Why not?

Well, what is delivered are legal codes and administrative organizations that promise order, predictable decisions, regularity of procedures, and responsible, objective and qualified officials; and an economics that operates according to principles of calculated advantage , efficiency and means-end strategies. Human action becomes a response to economic compulsion.

This kind of ratoinalization tendentially subjects institutions (the bureaucracy, hospitals, universities etc) to the norms of instrumental reason--efficiency, calculability, standardization.

Adorno's thesis is that it is the same conception of reason that governs scieintific rationality as governs societal rationalization.