Why can I use P = I²R but not P=V²/R when calculating energy lost in a circuit?
The problem assumes you understand something that is not clearly spelled out: the wires and the (unknown) load are in series. Therefore they share the current, not the voltage of the battery. 

That's the situation:







simulate this circuit &ndash; Schematic created using CircuitLab

As other have pointed out, the voltage drop across the wires is small given their small resistance. 

What you know is that the same current flows both in the load and in the wires, hence that's the information you must use to calculate power lost in the wiring.