THIS Corinna = Haec Corinna ( < hic, haec, hoc: this, pl. these).
THAT Corinna = Illa Corinna (< ille, illa, illud: that, pl. those).
THAT (awful) Corinna (that you like so much) = Ista Corinna (< iste, ista, istud: that (of yours), that (close to you); pl. those).
There's also the "weak" demonstrative, is, ea, id (this/that; pl., these/those, with not the specificity of the others): Ea Corinna.
The form of the word "that" that's used here (Illa) has to be modifying Corinna , and "pointing her out": "That Corinna (over there)", or "That famous Corinna."
You could use an accusative form of "that," to mean "that (thing)," as in, "She sings that sweetly." I would suggest the neuter singular accusative form, which is illud , for "that (thing)."
Corinna illud suāviter cantat .