That's a hard one. I'll list here what I can say about it so far, but I don't have any single definite source.
You provided the relevant etymonline link for the English etymology. I will add that dictionaries list gage as a possible alternative spelling of gauge.
The Middle English gauge comes from the Old French gauge (n.) /gauger (v.), which correspond to the Modern French jauge / jauger. In turn, none of my French dictionaries has a definite etymology for gauge. The Littré says (translated and heavily summarized):
Could come from Latin aequalificare or qualificare. Definitely related to (and influenced by) the Old French jale/jalaie (wooden measuring pail) and gallon (), which themselves come from a series of Late Latin roots including galida (from Latin galletum). Also related to the German eichen.
So, it is seen that gauge, at the time it was imported from Old French into Middle English, coexisted with a lot of words of similar meaning and close spelling. Thus, probably gauge took its writing from gauge and its pronunciation from a mixture of those words (gauge, jale, gallon).
Regarding the issue of whether the initial consonant is a soft or hard g, it is funny enough to note that while the English word, with it hard g, comes from the Old French (which had soft g), the Modern French uses gauge as a nautical term, imported from the English, with its hard initial g.