You should revise the articular infinitive on p 79.
‘Articular infinitive. The substantival force and case usage of an infinitive used as a noun are sometimes marked more strongly by the use of the neuter singular definite article to introduce the infinitive phrase. The article must be used when the infinitive functions as a substantive in the genitive or dative or as the object of a preposition. In the nominative and many uses of the accusative, either the articular infinitive or the bare infinitive (as exemplified in §§2a and 2b above) is allowed.
nom. τὸ ἄρχειν πόνον φέρει. To rule brings toil.
gen. ἐκ τοῦ φεύγειν as a result of fleeing
dat. πρὸς τῷ δῶρα λαμβάνειν in addition to taking bribes
acc. πρὸς τὸ ἐλαύνειν τὰς ἵππους with regard to driving the mares”
The articular infinitive is a syntactical unit and it can be declined. The examples M. Gives above are with a preposition. But you know the meaning of the dative case on its own. So ask yourself how would you normally translate a dative. You can give the type of dative a name if you like.
I guess I’d say that in “by the fact”, “the fact” refers to the articular infinitive. It’s a way of translating the articular infinitive into English. In the sentence, “The fact that your fled is a sign that you are a coward”, “The fact that you fled” is the subject of the sentence.
It’s the word “by” that suggests dative-instrumental to me because it is the means by which you achieved something. Fleeing is the means by which you made the battle bitter.
I think what is tripping you up is using the English translation to work out what the Greek means. The literal meaning of τῷ φευγεῖν is not “By the fact that you fled” but it is “by the fact of fleeing”. No subject is explicitly mentioned in Greek but it can be inferred from “ἐποiήσατε”.
If you had been asked to translate “By the fact of fleeing, fellow soldiers, you made the battle bitter for yourselves and sweet for the enemy.” The Greek would have been the same because it is clear from the sense of the English that the “fellow soldiers” are the ones who fled. In Greek articular infinitives are regularly expressed without a subject. See Cambridge Grammar of Classical Greek 51.40 p 602.
So at the risk of labouring the point when you read “you” in “by the fact that you” that “you” does not explicitly appear in the articular infinitive in Greek. In these circumstances when a subject is mentioned in English it is not expressed in Greek if it can be clearly inferred. Not every word in English is translated into Greek.
The dative here could be instrumental or causal see CGCG 51.46 p 604. I was not disagreeing with Mark it is just that I don’t get too worked up about what we call particular uses of the dative.