It doesn’t sound like they are wrestling. The teacher should have been fired the day the school learned that she was gay. Same thing if she were a hetero fornicator.
I'm grateful they did the right thing, quite possibly with pressure from Abp. Chaput. The school is not part of the Diocesan system but rather a [private school run by the Sisters of Mercy --- and thus does not "exactly" "directly" answer to the Abp, although he does have the ultimate authority (which Bishops too rarely use) and if he wanted to, he could revoke their right to call themselves "Catholic," and if push really came to shove, he could oust the Sisters of Mercy.
That would be considered the nuke level of Bishop authority.