As a counselor or other psychology professional, how do you mentally handle the stress of being unable to help a patient?
It is not easy. You engage in lots of self care. You get counseling yourself and you utilize supervision! You learn to accept that you have done what you can and you learn to let go. I suggest discussing this sort of thing with your supervisor (who you will have regular contact with once you start seeing clients). You have to also learn how to accept that you do not have a responsibility to "fix" people or to do the work for them...you do your best while you are with them, you learn from mistakes you might make and you move forward. The fact you are thinking about this now means you are a very caring and thoughtful person. I am very similar and it took me a little while to learn how to practice self care and how to accept that I am just a human too. There are going to be bad days and clients you think you haven't helped but I think the good definitely outweigh the bad.
Another thing to think about is that you don't know how you might have helped a client because they have a completely different perspective of "help" and of your working relationship with them. I would suggest reading "The Gift of Therapy" by Irvin Yalom: He has a great story in there about how he and a client both write a letter about their working relationship and their experience together in therapy. This story really helped me form a new understanding of the concept of "helping".