I'm sorry you've had to watch your daughter struggling to breathe over the past few weeks. As an outsider, it's horrible to read it and imagine. To have actually lived through that with your own child is distressing, to the point of being traumatic. I don't mean you have PTSD. But trauma is a form of huge stress about something you can't control, usually related to life/death in some way. So given that, it's perfectly normal for you to be exhausted, even without the lack of sleep.
Getting as much sleep as possible is important. So is taking time for yourself. If this means kids in front of TV more than ideal amounts for a whole, go for it. Anything else you find relaxing should be a priority too. The goal is to calm your nervous system down. It's accumulative too, so, for example, 20 minutes undisturbed to have a cup of tea is better than not.
Generally, sleep, fresh air, taking things a bit slower, cutting yourself some slack (you've been through a lot), moderate exercise like going for a walk, yoga, breathing exercises (plenty of apps with them) are all things that help calm the nervous system. However, I've found I don't like all of those, so I do what makes me feel calm. Could be knitting. Anything that you like to do basically.
It's a case here of putting in your own life jacket first. And it's not going to be forever.
Re your toddler, a bit of love bombing there will likely help. Hard to do when you've nothing left to give, but spending some one-on-one time with her, focussing on her, then (for example) she goes to GPs and you have something planned to do with her when she's back, telling her you're having a sleep while she's away (so she knows you'll be there plus not doing anything interesting!), might help? She needs lots of reassurance and your attention, no doubt along with some predictability/certainty. Her little life has also been thrown up in the air.
If the hospital think it's asthma, is there something they can do to help her preventatively? I have no idea about asthma treatments above inhalers.
I'm sorry you're having to go through this. It's bad enough when something happens to our children, but the aftermath of managing our own feelings, fears and ability to function on top of managing other children we may have is absolutely massive.