。
Ki was so happy she could barely speak.
Literal
Ki [topic-は] joy [possessive-の] too-much [almost-ほとんど] mouth [subject-が] could-not-speak.
木 here is almost certainly a person's name (a short Japanese given name, read き) rather than the word for 'tree' — a tree can't 'barely speak with joy.' うれしさのあまり ('out of too much joy') uses the abstract noun うれしさ (the -さ noun form of 嬉しい) plus ~のあまり, the fixed frame '(driven by) too much of X.' 口がきけない ('to be unable to speak') is the standard idiom, literally 'the mouth cannot function.'