1) "This case will show [how widespread has been the reputation of this man]."
It seems, at first blush, that the expression "how widespread has been the reputation of this man" in your example might be an exclamative content clause, which is functioning as the complement for the verb "show".
The inversion that you noted seems to be due to the exclamative's subject getting postposed (with the resulting overall effect similar to subject-dependent inversion). The version without that inversion would be:
- 2) "This case will show [ [how widespread] [the reputation of this man] has been ]."
Notice that version #2 has the exclamative phrase ("how widespread") fronted, which is what happens when it is not functioning as the subject. Aside: subject-auxiliary inversion does not usually occur for exclamative clauses when it is a main clause (and supposedly never if it is a subordinate clause) -- though it is available in main clauses it only occurs infrequently, or in literary style.
(Note that the word "how" is also used for interrogative clauses and fused relatives. The content clause in version #2 might be ambiguous as to its interpretation, as it could probably support both exclamative and interrogative interpretations.)
The 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL), page 921:
It is also possible for the subject to be postposed when the exclamative phrase is an adjectival predicative:
-  [How great] would have been [her disappointment] if she had known what they had actually thought!
CGEL page 991:
6 Exclamative content clauses
Subordinate exclamatives, like main clause exclamatives, are marked by one or the other of the exclamative words what and how:
One difference between subordinate and main exclamatives is that subject-auxiliary inversion is restricted to the latter, but since it applies there very rarely this is a minor difference -- quite unlike that holding between subordinate and main clause open interrogatives. As in main clauses, the subject may occur in postposed position: They emphasized [ [how imperative] had been [the need to take immediate action] ].