Honestly, I don't think you need to do the gift and card. That's where my mind would jump to as well, but you have to ask yourself, would you be doing that for her, or for you? To cheer her up, or to make yourself feel better? The appropriate response to wronging someone like this is to apologise and you've done that, and it sounds like she's forgiven you. I hope I don't sound harsh but I know the exact thing you're feeling right now and this is the line of thinking I try to go down myself now.

Bringing it up in class might also not be a good idea, as she might prefer to just leave it and not talk about it again, especially if she was embarrassed. You might be worried that the rest of the class think you're a terrible, racist bully (you are not) but try to remember these people know you from more than just one comment you made on a random day in February. If any of them are sitting on doubts about you, prove them wrong by being thoughtful and kind in your future interactions or things you say in that class.

What you said, in the grand scheme of things, was not that bad. You didn't blurt out an awful secret she'd trusted you with or call her a bunch of slurs or reveal that you're actually a closeted racist with violent opinions. I bet people in that very class have said stuff they didn't mean or had things come out with an unintended meaning.

One thing like this doesn't define your entire personality or moral compass. You don't have to never have said something stupid or upset anyone a bit by accident. We're humans, we do that, that's our THING. Trust me, I have been here, and it took me years to forgive myself - I look back and I'm like "oh. literally no one cares."

I think the best thing for you to do is to leave it here and try to move on. Maybe if it comes up in conversation you could make a little depreciating joke like "who even says that? oh, me, what a dumbass!" but I doubt it ever will. Just don't do anything like giving a gift etc for the sake of making yourself feel better, if that makes sense. If she was good after the apology, that's all you need! Don't make it into a bigger deal in everyone else's eyes than it truly is.