OP: so sorry about your DC. It does sound to me like you need time off to cope with the current situation, both from the point of view of looking after DC and managing your stress.
It is such a shame that there is no government recognition of the need for parents to have access to paid leave in this kind of situation. (It is a big step forward that there are now proposals for paid leave for parents whose newborns require neonatal in-patient care although I am not sure whether the employer or the government is going to pick up the bill for this. As someone whose baby spent eight weeks in SCBU, I well understand the financial implications currently for mums and partners of having a newborn in hospital for an extended length of time.
As an owner of a very small business which used to employ staff (in addition to myself), I just wanted to pick up on a statement made by a PP saying that the government reimburses companies for sick pay. This used to be the case but sick pay is now paid for by the employer with no reimbursement. For a very small company like mine, a member of staff going off on extended sick leave would be very bad news financially, and it is one of the reasons I decided a few years ago that the risk of this, plus other changes in the law leading to increased staffing costs, was too great and that I would stop employing staff and instead work at a level which I can cover myself, working with freelance colleagues if I need help for temporary projects. I didn't make staff redundant but scaled back gradually as staff left and did not fill vacant posts.
Larger companies are likely to be able to cover the costs of extended sickness absence as it will be a much lower proportion of their overall costs. I assume that many of the managers posting above in support of OP are employed in large companies where the issue is more of re-allocating work rather than cost.
I would really like to see the government addressing this issue as there must be many parents who find themselves in a similar situation to OP, and some of them will be working for very small businesses who have to pick up the tab.
OP, this post is NOT directed at you. You are in this situation now and you need to do what is right for you and your family.
It's more to highlight that this sort of situation is an issue for small businesses and to say that the government should be ready to at least provide some help towards the costs. Small businesses are a major employer collectively and if the government wants to see economic growth (including increased employment), it needs to realise that it needs to support the small business sector better than it does at present.