My mother was humming a song while busily cooking in the kitchen.
Literal
Mother [topic-は] kitchen [at-で] diligently cooking-[while-ながら] song [object-を] was-humming
せっせと = diligently/busily — a mimetic adverb for steady, industrious work. ~ながら marks simultaneous actions: cooking while humming. 口ずさむ = to hum (口 mouth + ずさむ to mumble/sing softly). A warm, evocative domestic scene.