。
She drilled the basics of English grammar into her students thoroughly.
Literal
She [topic-は] students [recipient-に] English-grammar [possessive-の] basics [object-を] thoroughly teach-drove.
教え込む ('teach + drive into') is a compound: 教える ('teach') + 込む ('go into / inside / fully'). The 込む suffix carries the sense of the action being driven inward, emphatically and completely — much stronger than plain 教える. 徹底的に ('thoroughly') is an intensifier from 徹底 ('thorough, exhaustive') + 的 (suffix making nouns into adjectives) + に (adverbial). Layered together, the sentence depicts no-quarter teaching: drilled in, hammered home. Common verbs that pair with 込む in this 'drive in' sense: 押し込む ('push in'), 信じ込む ('believe firmly'), 巻き込む ('drag into'). The 込む family is one of the most productive compound-verb suffixes in Japanese.