I'm sorry, what an upsetting situation. Unfortunately while it is great when employers will go above the law, they don't have to and in the circumstances you describe if you have no paid leave left and under your company's policy they do not pay for your statutory parental leave (which most don't) then there is nothing else they have to do.
What are your usual childcare arrangements when you are at work, and how long until your DC is well enough for them?
Can you use the two days paid leave to make arrangements: talk to the hospital discharge people about your difficulty and see if they can suggest anything; find a local babysitter or emergency childminder who can cover; ask all your friends if anyone can help you out, even for the odd day?
Depending on how old your child is and how much looking after they need, do they need someone all day every day or can you take eg three days unpaid leave and the other two days go to work and leave them with the phone, a sandwich made, TV remote and a neighbour pop in at lunchtime to check on them?
Do you do a job where you can work from home, so that you are there if you are needed but can get on while your child sleeps/watches TV/rests?
I hope you find a solution.