Kirk hypothesizes that high pressure is the Zetarians' weakness; and he gambles that it is the only hope to save Mira. The pressure reached a point were the Zetarians were in so much pain, even
debilitating pain, they couldn't stay focused on staying in Mira's body, nor mustering the strength to phase through the chamber wall. They simply oozed out of her and died. Kirk wins again.
As for killing Mira with high pressure, as a former scuba diver who dived to 110 feet at about 4.2 atmospheres of pressure (1 atm. of air plus 3.2 atm. of water) and felt fine at that pressure, the human body can sustain multiple atmospheres of pressure. Since she was breathing compressed air, this is akin to scuba diving and not free diving (no scuba). Wiki indicates that people start to die around the 300 foot depth level on compressed air. That's about 10 atmospheres or about 150 psi. With 23rd century technology, maybe this could be higher. As
@TLOZ Lover points out in his blog, "the bourdon gauge on the test chamber wall ranges from 0 to a maximum of 5.0 (atmospheres), and it stops about 3.30 at the point when the Zetarians are finally driven out...". If 3.3, then no big deal for Mira. Even if maxed out at 5.0 atmospheres, still no big deal for Mira.