Chances are good that she likes you, too.
Usually, women signal interest more subtly by causing them and their interest in question to spend time together, even if by no means romantically. And since she invited you to her exercise, it sounds a lot like she wants to spend time with you.
Of course, I am an outsider and can't say for sure, but I wanted to add that for reassurance.
You stated two distinct goals:
Approaching her without creating an awkward scenario and not being too direct/bashful as to cause her to go to defense and making excuses/delaying. (I find my description of the latter lacking, but you'll see what I mean.)
She has made clear through her actions that she likes spending time with you. Sou you should use that. In your case, as you do not want to risk your friendship and dance partnership, you might not want to be too direct.
Invite her to things friends do together. When you go out with your friends, maybe invite her, too. (Ask your friends about that, first, though.) Friends can do a lot to make someone look more attractive as group scenarios and dynamics do show your social skills as well as friends would usually try to make you look good or fun to be with, too.
Another thing you could do is invite her to dinner. I would recommend a restaurant and not your place, but that depends on cultural context and personalities (and your cooking skills). It is rather obvious in terms of where you are going with that, but it is not nearly as direct as "I want to be with you" or something of that sort.
Stating it like this would give her the option of rejecting a dinner invitation instead of you. So even her rejection would be indirect.
For this it might be best to not give a timeframe at all or a vague one. Otherwise you could barely tell a rejection from an actual excuse.
There is also the alternative of getting dinner tonight spontaneously. Maybe after dance class you ask her:
Hey, I wanted to grab dinner at [not-too-fancy-restaurant of your choice], want to join me?
It would not give her the opportunity to clearly signal "I like you, but I am not interested in dating you." (if that was the case), but you could test the waters in regards of how much she seems to be interested in spending time alone with you, in a very roughly date-like scenario.
You did not want to be direct, so you should try to bond on the basis of friendship. This approach might not give you a direct answer of "yes" or "no" within a single conversation or even day, but it is a lot less likely to make it awkward for you to still spend time together.
Have fun together, enjoy activities together if she enjoys all that with you, chances are good you could make it work. If you have trouble looking for positive signals of "I'm into you", consider looking for negative ones of "I'm not comfortable with this". This does not mean you should try things you think could make her uncomfortable, but to see her reactions on subtle stuff. Like simple, random body contact. If you accidentally touched her hand, does she pull hers back intentionally or is she just surprised. None of these things are very clear signals, but you'll get an overall feeling with this.
If you get the right feeling, either change the subject to relationships to find out if she's seeing someone at the moment, or ask her out to something quite obviously date-like. Especially with the latter you will get a rather clear yes or no. Phrase it as an invitation and not a "I want to be with you". That way it is more likely she is going to be honest, if she intends to reject you, but avoiding being direct about dating gives her more space and opportunity to be honest instead of cautious.
I also want to make it is clear as possible that I do not mind it if she rejects me [...]
Phrasing is very important. You should certainly not have her see it as "you do not mind if she rejects you", because if that was the case how could she be interested in you if you cared that little?
What you should signal is that you could bear rejection, not that you don't mind.