Lots of us have been in similar situations, of course we're behind you!
Of others have said, your DF will not want you to shower him, if he does need help with showering, it's much better done by a Carer.
Also, it's not up to your DM to volunteer you.
If your DF has to stay in Hospital longer whilst a Care Package is put in place, then that's just how these things work. Please do not feel guilty.
All of the advice above is really good. I'd just like you to add, please do call the Discharge Clerk on his Ward tomorrow and ask what the plans are for Discharge for your DF. Specifically ask if he's had an Occupational Health Assessment and whether they'll be assessing his home and aid he's had a Physio Assessment. Also tell them that you won't be providing any personal care, like washing or toileting.
I'd also ask if he has a SW. If he does, ring them and tell the SW that you won't be providing personal care for DF.
If your DPs don't have them already, I'd recommend a cleaner and a gardener and possibly a milkman. As well as helping them manage to stay in their own home, they're also extra people to keep an eye on them.
And one final thing. Have a look at the book, The Twilight Shepherd on Kindle. It's very simple practical advice for those of us with DPs who are aging and need help.