Does your employer have an employee assistance programme or similar? I've had it with two employers (big multinationals, which may well make a difference) - it's an advice line, with access to all sorts of different advice, and the only information you have to provide is your employer (they get stats, e.g. 10 employees are having personal counselling, 3 have had financial advice, 4 have had legal advice - but no identifiable information.) If you've got that, it's a service free to you, so use it.
Is there anyone else you can talk to? I have been having a challenging year (it got to the point where I walked out of the office one afternoon, because the way things were going, it was making me feel like sticking sharp things in my wrist), and after one meeting I was in, someone commented that I didn't look as if I was on my best form, and did I want to talk about it (shes from an entirely different department.) I said no (I am my own worst enemy) - but later mailed her, and we then had a separate 1-2-1, and she just helped me see some of the wood for the trees, which things can I do something about, which things are without my control. It's just been so good to have someone on my side, who will listen, and has given me some hints, but mostly just shows me she believes I can make things better myself. And I shouldn't have waited - she's someone I could have gone to anyway, but I am very good at putting my own needs last, and she's always so busy, and... So don't do the same as I do!
Have HR been able to give you any advice about how to handle things, as you've been to them about it?
Sleep deprivation does make everything a million times harder and more difficult to cope with - it's no coincidence it's used as a form of torture. Is there anything you can do to catch up a bit? I'm not sure how old your DC(s) is(are) - could exH take them for a weekend, or grandparents, so you can really relax and sleep in and not do housework or catch up on anything else, but focus on sleep?
Bumblebee's point about delegating more and getting others to step up is good, too - it's good for them, as well, if it will stretch them a bit and gives them a chance to try things they might not always do and may be a little outside their comfort zone.
As for the staff member - keep all the mails. Each on their own may not mean much, but they can help build up an overall picture if things step up. I would stick to giving her a fair report based on evidence, as you're planning to do. Does your employer have guidelines on how to write and conduct reports? It may be worth reviewing them - my manager clearly hasn't read ours recently, as it states you shouldn't give mixed messages, and I had quite a few positive remarks that were followed with, "but..." which just diminishes them, makes it feel like they don't count for anything. So avoid doing things like that, and separate positive and negative comments. Even if your employer doesn't have guidelines, there's loads of stuff online about how to give feedback. There's no harm in reviewing things like that when you know you're struggling to think clearly through exhaustion.
If she's on sick leave, is it also worth sending a one-off reply to say that you appreciate her keeping in touch even if it's a lie, but you want her to concentrate on herself and her health, and she needn't mail you unless it's to send in a fit note or discuss coming back to work? It would need wording carefully, and it may not be the way to go at all. Perhaps it's another thing HR could advise on?
I have done a Franklin Covey 7 Habits course - does your company offer it at all? A lot of it seemed like common sense to me, but I'm actually quite good at some things like time management, and one of my weaknesses is that I struggle to understand why other people can be so bad at it. My main issue with it was that it's all very well going after your goals and sharpening the saw and so on - but I don't have much in the way of goals and direction, except very short term ones, and I didn't feel there was much to help me identify what my direction should be before I can set about achieving it. But it's still useful if you can do it. I particularly liked the quadrant for prioritising tasks, and there's also some useful stuff about communicating with others and considering their point of view which might be helpful in this situation. Probably any library would have the book(s - they've branched out in a number of areas).
Good luck with everything, and especially getting more sleep, as that makes so much different to everything.