。
I entertained her family lavishly.
Literal
I [topic-は] her [attributive-の] family [object-を] greatly entertained-[insistence-よ].
大いに ('greatly, very much, heartily') is a formal/literary adverb. もてなす ('to entertain, treat, serve') is the core verb behind the Japanese cultural concept of おもてなし ('hospitality'). Used for going out of one's way to welcome and serve guests.