。
The girl reading the cookbook is Kate.
Literal
Cooking [possessive-の] book [object-を] is-reading girl [topic-は] Kate is.
A relative clause (料理の本を読んでいる) modifies 女の子 — Japanese relative clauses precede the noun they modify with no relative pronoun needed. ~ている marks the ongoing action of reading. The whole sentence is an identifying copula structure: [described person]は[name]です.