OVER-using adverbs is a sign of bad writing, just like OVER-using anything else.
In your first example, I think it's fine. There are different kinds of smiles, it's hard to describe the differences between them using "showing" words, and it's important that people know what kind of smile this is. So I don't have a problem with "kindly". Technically, I guess you could let people assume she's being kind since her words are kind, but there are different ways to say those words, too. The "kindly" makes it clear she's not be sarcastic or superior with the words... I'd leave it.
The second one is a bit dodgy, just because it's redundant. "Apologetically" and "as if he were personally responsible" give pretty much the same information. I'd pick one or the other.
So, while some writing "experts" promote a near-complete ban on adverbs, a lot of these experts judiciously use adverbs themselves. (I'm looking at you, Stephen King). Like everything else, they can be over-done. But that doesn't mean they aren't useful in their place.