I'm going to be direct with my significant other about her threats of suicide.
Oh no, is there a reason she's threatening suicide?
She's very unhappy and I'm worried about her.
You have to talk to her and tell her how worried you are about her mental state. It's not acceptable for her to threaten suicide and not get help. She's putting you in a bad situation and affecting your mental state which is not healthy for either one of you.
I guess you're right. I'm sorry for being so angry.
It's understandable that you're upset; she's putting you in a difficult spot here. Be there for her, gently, and try to get her to call a hotline or see a therapist.