I was busy, so I didn't write a reply to your letter.
Literal
Busy-was [because-ので] your letter [possessive-の] reply [object-を] didn't-write.
ので gives a reason in a softer, less subjective way than から — it presents the busyness as an objective circumstance rather than a personal excuse. の stacking: 君の手紙の返事 chains two の particles to build 'the reply to your letter.' 書かなかった is simple past negative.