IPA: /ˌɪnɪkˈsprɛsəblɪ/
KK: /ɪnˈɛkspresəblɪ/
In a way that cannot be expressed or described in words.
She felt inexpressibly happy when she received the good news.
Inexpressibly → It is formed from "in-" (meaning not), "express" (from Latin "expressus", meaning to press out or convey), and "-ly" (meaning in a manner). The word means in a manner that cannot be expressed or conveyed.
Think of 'not' ('in-') being able to 'convey' something ('express') — that's why inexpressibly means something that cannot be expressed.