。
She asked him to open the window.
Literal
She [topic-は] him [to-に] window [object-を] open [in-order-to-ように] asked.
The ~ように頼む pattern ('ask (someone) to do X') uses ように to attach a request as the purpose of the asking. The に on 彼 marks the recipient of the request. Compare with the simpler quotative ~と頼む (preserves the request verbatim) — ように頼む reports the request indirectly as a desired outcome. Both are common; ように feels slightly more polite/indirect.