I think there are actually 2 different issues here.
The first one, and the one that you really want to let off steam about is that your DH is only focusing on work but has checked out of any other family relationships right now.
I agree there isn't much you can do about this for the moment. My advice would be to give him space and time to come to terms with things for now. Hopefully once he has better treatment and has had time to adjust, you will be able to address this with him. I am really sorry that this is happening and I can see how this makes you feel everything is on you.
The second issue is the practical one of him working at random hours in the lounge of a house which is fairly open plan and with two young children, one of whom at least is so young that they aren't going to really understand why daddy isn't responding.
This is what I think you might address before the summer holiday. However ill and in pain he is, you can't have him taking up that space with two young children at home needing to play and have fun. Their emotional and developmental needs are also important.
Whatever you do, you are going to feel like the "single parent" for now as he doesn't have the bandwidth to do any different. Yes, this is hugely unfair to you when you have also been coming to terms with things.
Find a place to vent and let off steam about it ( does your work offer employee counselling, are there any local carers groups that might help? ) but also you do need to find another practical solution. If you feel like the single parent now, will it really make much difference if he is working in another room and still detached from everything?
I understand that the chair is expensive. (Would his work be amenable to finding a second one, or at least part funding one,?)
You need to find a way to get him out of the lounge. You need to bite the bullet and tell him it just won't work during the summer holiday
I don't know the layout of your house but you need to find the least worst option that works.
This would be my advice.
Once he isn't impacting everyone to the same extent it will be easier to manage being the ( hopefully temporary) single parent.
I am so sorry this is such a difficult time for you op.