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She not only manages the household, she also works as a school teacher.
Literal
She [topic-は] housework [object-を] manage [not-only-だけでなく] school [of-の] teacher also is-doing.
Two pieces give this sentence its texture. 切り盛りする (literally 'cutting and serving,' as in dividing and dishing out a meal) is the verb of choice for managing or running a household, shop, or operation — keeping all the moving parts coordinated. ~だけでなく ('not only X but also Y') is the standard 'not only X but also Y' marker, slightly more formal than the spoken ~ばかりでなく. The closing もしている ('also doing') uses も to add the second item ('school teacher') to the list. Cumulatively the sentence praises someone juggling two demanding roles.