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I'll come home in time for mother's birthday.
Literal
Mother [gen-の] birthday [to-に] in-time-be [so that-ように] return (polite).
Two useful patterns fused. 間に合う = 'be in time for,' idiomatic and very common around schedules and deadlines; it takes に to mark the thing one is meeting. ~ように attached to a verb expresses 'so that X happens' — a purposive link between 'be in time' and the returning. Together: 'return so as to be in time for mother's birthday.' The subject (the speaker) is omitted.