She has a charming face.

Literal

[topic-は] charm [subject-の] exist face [object-を] is-doing.

愛敬 is an alternative spelling of 愛嬌 ('charm, amiability') — the kanji 敬 ('respect') in this older form gives a slightly different flavor, evoking warmth tinged with respectability, though in modern use the two are largely interchangeable. The structure 愛敬のある + 顔 is a relative clause modifying 顔 ('face'): 'a face that has charm.' Crucially, の inside a noun-modifying clause can substitute for が as the subject marker — this is a common feature of relative-clause subjects. 顔をしている ('is doing a face') is the standard Japanese way to say 'has a [type of] face/expression,' similar to 元気な顔をしている ('looks well') or 怖い顔をしている ('has a scary expression').