My mother is busy cooking, doing laundry, and so on all day.

Literal

Mother [topic-は] all-day cooking did-[alternating-たり], laundry did-[alternating-たり] do [nominalizer-の] [with-に] busy

~たり~たりする = 'does things like X and Y (among others).' Lists actions non-exhaustively. 一日じゅう = all day long. The pattern conveys the mother's endless rotation of household tasks.