Italy was the center of Roman culture, and Roman institutions in great part carry over into Theodoric's Kingdom of Italy mostly intact. We can literally consult the writings of the guy who was effectively King Theodoric's Prime Minister, Cassiodorus. The Lombard Dukes also seamlessly integrated into the Roman Senatorial class fairly rapidly, leaving numerous latin records. In following periods, ecclesiastical authorities engulfed preexisting Roman institutions (this part of the reason why Churches that have been granted special privileges were —and still are— called "Basilicas" just like a kind of Roman Courthouse).

Also, it's a bit of a misconception that there aren't any written records in the Middle Ages. The main issue is that records and chroniclers generally have to do with Political History and don't tell us much about how people in the period actually lived. In essence, the Middle Ages were Europe's transition from a sprawling empire to compact, ethnically and geographically defined polities (i.e. the predecessors of modern Nation States). Thus, it is understandable that this transition would be chaotic and chronicles from the era are just endless narratives of people fighting, betraying, and murdering each other. But make no mistake, there are chroniclers: Marius Aventicensis and Gregory of Tours (both VI century Burgundians), Fredegar and Paul the Deacon (VII century, a Burgundian and a Venetian) and Liutprand of Cremona (X century, Italian). These authors can be consulted and cross-referenced. There are also ecclesiastical chronicles (the "Liber Pontificalis") as well as purely Frankish sources (the "Annales Regni Francorum") in addition to legal and administrative records. There was a conversation about this here three months ago you might be interested in.