I'm not sure the Borg have ever demonstrated the ability to fight with fragments of a ship, or even with a seriously damaged ship. A swarm of Cubes will probably operate like a swarm of Drones, with individual units taking great risks because even their destruction will be to the advantage of the Collective. But individual ships have been disabled and negated as threats in many episodes, starting with "Q Who?", until time-consuming repairs could be effected - and extensive damage has resulted in infinitely long repair times, such as with "Unity".
As for the Borg sustaining an attack until the opponent is subdued, it's something we only hear spoken of. Our onscreen heroes have never quite had to face that yet - except perhaps in the long, strategic term. What rather happens to them is that they conduct a strike against a defensive Borg force, and attain their limited tactical goals even though the Borg defenses are not significantly weakened for the loss of half a dozen Drones or some Cube systems. When the aims are limited, a limited spectrum of "surprise" weapons is a valid way to attain them.