Learn English – What’s the “A fine thing it would be” refering to in this context

"I know that," said Professor McGonagall irritably. "But that's no reason to lose our heads. People are being downright careless, out on the streets in broad daylight, not even dressed in Muggle clothes, swapping rumors."

She threw a sharp, sideways glance at Dumbledore here, as though
hoping he was going to tell her something, but he didn't, so she went
on. "A fine thing it would be if, on the very day You-Know-Who seems
to have disappeared at last, the Muggles found out about us all. I
suppose he really has gone, Dumbledore?"

This is from the book Harry Potter. I'm confused by the sentence in bold. I don't know what "A fine thing" here is really referring to? Is it "on the very day You-Know-Who seems to have disappeared at last" or "the Muggles found out about us all"?

I am not sure how to understand the sentence correctly?

My best attempt is:

It would be a fine thing if the Muggles found out about us all, and on the very day You-Know-Who seems to have disappeared at last.

But it sounds like two fine things, not a fine thing. And it seems to me that it's an ironical sentence.