is she a career/school/guidance counselor, or a therapist?
I think it's ok to open up about the dark stuff to your therapist. I mean, the other stuff can be helpful, but only if you are in a place to be able to address it. If you are constantly consumed by the sui ideation and depression, then that's what you need to focus on. You can always come back to scheduling your days and getting career advice.
My wife's therapist actually hooked her up with an employment specialist (not totally sure what her job title is, but she helps my wife look for work and figure out more of what she wants to do in life)... If you are talking to a therapist or mental health counselor right now (as opposed to a life coach or career counselor), then there's no reason not to talk about the depression.