None of these was the version I was looking for, but they’re all perfect in entirely different ways. (All youtubery.)
Diana Krall, “Exactly Like You” (includes “Let’s Face the Music and Dance”)
Django Reinhardt & Stéphane Grappelli, “Exactly Like You” (audio only; closest, but no singing)
Tom Jones with Eric Morecambe and Ernie Wise, “Exactly Like You” (incredibly silly)
“Exactly Like You”
Lyrics by: Dorothy Fields
Music by: Jimmy McHugh
From Lew Leslie’s International Revue (1930)