。
She showed off her English skills to the fullest at the speech contest.
Literal
She [topic-は] English [genitive-の] speech-contest [at-で] without-restriction that talent [object-を] demonstrated.
Three high-register pieces interlock here. 遺憾なく (often kana-spelled いかんなく) literally means 'without 遺憾 / without regret,' i.e. fully, unrestrainedly — 'leaving nothing wanting.' 発揮する ('to manifest / demonstrate') pairs almost exclusively with 力, 才能, or 実力, and the pairing いかんなく~を発揮する is so set it has the feel of a stock phrase, common in school awards announcements and sports commentary. その才能 ('that talent of hers') uses その in its anaphoric sense: pointing back at her English skill mentioned earlier in the implicit context. スピーチコンテスト is a common loanword combining English roots — Japanese readily strings together English borrowings to name modern formats like this.