。
She blew her nose into the handkerchief.
Literal
She [topic-は] handkerchief [with-で] nose [object-を] blew.
鼻をかむ ('to blow one's nose') is the standard idiom — かむ means 'to bite, chew' but in this fixed phrase it describes the squeezing-out motion. The kanji 擤む for this specific 'blow one's nose' sense is rare and almost always replaced by kana かむ, especially since the 'bite' kanji 噛む would be confusing here. で marks the handkerchief as the means used. In Japan, blowing one's nose loudly or using paper tissues in formal social settings is considered rude; the cultural preference leans toward subtle nose-blowing, often done away from others.