Thanks for explaining more OP, I see a bit more why you are worried when you say the general demeanour of both her and her dh is just worn down, a bit disengaged and depressed really all the time
For this I would suggest a friend sought help from GP or saw a counsellor whatever the cause might be. If she had counselling she might be able to find out why she is not enjoying things. It matters not whether a kid has happy parents at home or happy parents at work but I do actually think it matters that they have happy parents, although I really mean coping and not depressed parents.
If there is stress from having a child with extra needs or a real sleep problem, then there are ways to take care of yourself to help limit the impact. Having been in that position myself there are periods where getting through the day is the best you can hope for though, but even so taking care of yourself and doing things that make you feel even a bit better are still beneficial. But if you are depressed or struggling it gets harder to prioritise these things.
I would not suggest counselling should or could help a person find a 'mothering instinct' (I don't really believe in it) but I think counselling could help someone find a better headspace generally. Having depressed parents is bad for kids.
Things like kids' clubs etc, that is normal for many people, lots of people consider a holiday to be not having to look after the kids.
The only other thing I wonder is the things she cancels are when it is all of you together. I used to do this when me and DH were going through a very bad patch with stress after one of the kids was ill, we would row and I couldn't slap on a happy face so we'd cancel going to parties or whatever and just focus on the kids and not rowing. Is there any chance this could be happening?
I don't think you can do too much, but talk to your friend gently about how she is feeling in herself and then encourage her to seek help if her feelings are too much, rather than getting to much into why IYSWIM.