At what point in history did the kings of Francia stop speaking the germanic frankish language and adopt Latin/Old French?
The short answer is that we don't really know, but it was likely around 842.
Joszef Herman states that Classical Latin ceased to be used as a daily language in the 7th century, so, though it was obviously a gradual transition, this can be considered roughly the birth of the Romance Languages, including the ancestor of Old French, though, because of Gaul's distance from Rome, it is likely safe to assume increased regional variation leading to an earlier transition than occurred on the Italian peninsula. Old French, as a Latin descendant, would have been used initially by the Roman cultural descendants in what became Francia, who were conquered by the Frankish-speaking Franks by the 530s. While there are no sources specifically describing the social transition from Roman to Frankish domination, at least none that spring to mind at the moment, there likely developed a scenario similar to that seen in the wake of the Norman Conquest of England, with the higher strata of society speaking one language (Frankish) and the lower speaking another (Latin or a Latin derivative). Because of the documents resulting from the Third Council of Tours, we know that Classical Latin was unintelligible to the inhabitants of "France" by 813.
The Franks were not a literary people, as a rule. In fact, one of the defining characteristics of the Merovingian period is the paucity of documents that have come down to us. Almost without exception (and without exception when speaking specifically of the court), however, those documents that were created / have survived are in Latin, which would remain the official language of the French court until 1539, when Francis I signed the Ordinance of Villers-Cotterêts. Add to this that Charlemagne was singled out for praise due to his didactic enthusiasm (though he had trouble with Latin), which gives a strong indication that his predecessors were functionally illiterate, and generating an accurate linguistic profile of the Frankish court prior to that time becomes virtually impossible.
As for the development of Old French, the first court-related written documents in the historical record written in what would later become what we think of as Old French are the Oaths of Strasbourg (842), only thirty years after Charlemagne's death. Interestingly, this document states that Charles the Bald spoke in German, while his brother Louis the German spoke in "romance." As the two were brothers, this difference in language was likely a gesture by the kings to the members of their courts, with each king speaking in the language of his brother's courtiers. Considering that the Treaty of Verdun, signed in the following year, divided the Frankish lands such that "France" was essentially created, it makes sense that these political borders would foster increased divergence between it and the Germanic-speaking regions from which it was separated, further bolstering the development of a Romance-dominated region. A century and a half later, the ascendancy of Hugh Capet and his descendants would further solidify the regional dialect of Paris as the most politically important vernacular in Northern France, eventually leading to the development of French as we know it. The literary dominance of Old French at the time of the 12th Century Renaissance indicates that the French court was certainly speaking Old French as its vernacular by this time.
Sources you may be interested in consulting include: