'Wrongheadedness' is the noun for the quality described by 'wrongheaded': thinking that is aimed in the wrong direction and stubbornly stays there.

It is uncountable, and it usually appears in a small set of shapes — 'the wrongheadedness of the policy', 'sheer wrongheadedness', 'a mixture of arrogance and wrongheadedness'. When you turn the criticism into a noun like this, you can name it as a thing and then talk about it: how big it is, where it came from, what it cost.

The word is long, formal, and clearly the voice of writing rather than speech — editorials, reviews, history books, serious essays. Nobody says it at dinner. Because it names a fault in how someone thinks (not just what they got wrong), it is a strong criticism, and writers often soften it with 'a certain' or aim it at a plan rather than a person: 'the wrongheadedness of the whole approach' is easier to publish than calling a named person wrongheaded.

If you feel the word is too heavy, plain English works: 'they were thinking about it in completely the wrong way'.