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She wants to pass that exam no matter what.
Literal
She [topic-は] that exam [target-に] no-matter-what pass-want-third-person-progressive.
何が何でも ('no matter what,' lit. 'whatever is whatever') is a fixed adverbial phrase expressing absolute determination — heavier and more emphatic than どうしても. 合格したがっている stacks several pieces: 合格する ('pass [exam]') + ~たがる (third-person desire suffix) + ている (progressive). Japanese reserves bare ~たい for the speaker's own desire; describing someone else's wanting requires ~たがる, since one cannot directly assert another person's interior state. The progressive ている here gives the sense of an ongoing, sustained want.