In U.S. English, the simplest way would be to ask
What time does she get off work?
A shade more formal would be
What time does she usually leave work?
or
What time does she usually leave for the day?
In the latter, the work context would be understood. Don't use it if there could be some confusion.
Here's one way to ask involving the word stay:
How late does she usually stay [at work]?
Be careful, though, because any of these might be construed by the listener as evidence that you may be nosy at least, and possibly a stalker. So unless you're checking up on this woman as a job applicant, the prudent course would be to ask her yourself.
addendum for OP
You asked where you should place the until in your constructions. Presumably you're wondering whether you should avoid putting it at the end of the sentence. The answer is: don't avoid that. While some pedants will wince at this, they really don't have a leg to stand on. See this accepted answer on ELU to the same question. (TL;DR: This is a rule made up hundreds of years ago by fussbudgets who determined, against all usage evidence to the contrary, that English should behave more like Latin.)
Therefore you should feel no qualms about asking
What time does she work till?
or
What time does she work until?
But since English is pretty flexible, you could also easily say
Until what time does she work?
But for my money, that sounds a little more formal, even stilted.
Note that till and until are pretty much interchangeable here. See this related discussion on till and until on ELU.
Best Answer
All are generally acceptable for a response. I find that my response changes based on how well I know the person or not.
If I am fairly familiar with them (close friend or good friend) I may say:
or used in rare occasions that you want to acknowledge and show you are happy that your words meant a lot to them:
If someone says it that I am not so familiar I tend to respond:
English is weird where there are many ways to say the same thing. Many responses are appropriate for the same thing. I would just say what you feel is a natural response.
I prefer "no worries", others may say "no problem". Pretty much the same meaning or implied meaning using a different word.
In the above example you provided, their response of "that means a lot" can be interpreted as saying:
So I would word your response as if you are responding to that :) In the end as I said, there are many ways to respond and all the above are equally appropriate. Find one that feels natural to you and go with it!