What you're training
An applied leading-tone chord is the other way to tonicize a chord — instead of borrowing that chord's dominant, you borrow its leading-tone chord. viio7/V and viiø7/V are both built on V's own leading tone, a half step below it, so neither one has a root that can fall a fifth home the way a dominant does. All the pull comes from that one half-step and the diminished intervals resolving inward. This drill trains you to tell that apart from a plain applied dominant by ear, and then to split the diminished chords themselves — half-diminished from fully diminished.
Two levels, one skill
Level 1 asks the flavor question first: was that resolution to V pulled by a dominant (V7/V) or a diminished chord (vii°7/V)? Same target chord, same register — the only thing that changes is whether there's a root a fifth below V or not. Level 2 raises the bar — every round now has a leading-tone chord, and you name whether it was half-diminished (viiø7/V, a minor 7th above the root) or fully diminished (vii°7/V, a diminished 7th above the root). One note is the entire difference, so listen for it, not for the shape.
Go deeper
This drill is Trio 2 of the chromatic-harmony series — the diminished cousin of secondary dominants. The theory: tonicization → covers why leading-tone chords tonicize the same way dominants do. Read it in a classic textbook — every example plays → For the real-world payoff, Find them in real pieces → surveys every viio7/x and viiø7/x in a 57-piece corpus — around 150 applied leading-tone chords in all, with viiø7/V the most common at 28 appearances, then viio7/V at 22 and viio7/v at 21 — playable and labeled in context. The dominant flavor → is where to start if V7/x is still new — train that pull first, then come back and split it from this one.
Where it fits
Comfortable with chord quality and cadence resolution already? Applied leading-tone chords are the second lane of tonicization — same destination chord, a different route to get there. The full drill ladder has everything else.