If there's a healthy rhizome in the soil then it should come back. Just cut it back and it'll regrow. It might depend, I guess, on how much vinegar soaked into the soil.

Thanks for the comments. I rinsed the plant in water twice yesterday and cut back what I thought needed. It looks like this now. Wondering if I needed to cut more because basically 90% of leaves have some level of blackness and I've cut off most leaves that are 50% black. It's now 1/2 of it's original size.

Well it already looks a lot better from what I can see. It's up to you how much you cut back; seriously, you can cut it all the way down to the soil level and it's still going to grow back.

I cannot stress enough that vinegar is used as a contact herbicide. I am curious as to where your wife heard about using vinegar as an insecticide. It's the second time in as many days that someone has wondered about using vinegar in this way.

It looks as if the plant is indoors? Next time you get white fly on an indoor plant just put it out in the next rain shower. In fact Lemon Balm would be happier and healthier if it was outside permanently. Indoors it is likely to be weak and sappy and easy prey for any passing bugs.