Oh goodness, a kitten dying was the last thing you needed! ☹️ But accidents happen, especially in a busy home of kids and pets. Try move on - you all handled it and coped.
I'm sorry the phone call wasn't a help - but I think these call handlers are trained to encourage you to do most of the talking, and that doesn't always come easy. But! You made the call, OP. And you can do it again. And I think you DEFINITELY need antidepressants as part of course of treatment. They're not for everyone, true. But they can turn things around. Keep reminding yourself that your thought processes are the result of a very physical condition. Keep your resolve about counselling.
You're not in the best place to cope well with separation, but perspective will come. If you're the partner of someone who suffers from depression, it's VERY difficult to understand, and to help. So I think you did a good thing for both of you. It really doesn't sound like she wants a clean break. She wants to see you healthy, and when that happens, dialogue will be different and better.
Your next task is to make a GP appointment, as an emergency - not for a week or two week's time. You tell them EXACTLY how you are feeling, and you don't leave the consultation until there's the beginnings of a plan in place. Write it down and hand it to the doc, or the receptionist, if you're finding it hard to talk.
Are you eating? Are you resting (if not sleeping)? Food is important, because you need some energy reserves when your mind is exhausted.
I think you want to get out of this situation, and I think you know the reasons why, so keep going. It might not feel like momentum, but it is. X