Exactly, her timings are all over the show, although, I get she has bigger things on her mind.
If they are 2.5hrs away, that's a commutable distance, many people commute that who work in London (my DH included).
But saying this as a mother who has had both of her babies admitted into paediatrics for week long stays each (I get your stay is much longer), my DH never stayed over. He continued to work as usual, and we spoke/text via phone to keep him up to date. I do understand your child is going through a difficult time, but it would be unusual to have both parents there all of the time.
If your DP isn't working, then he can commute in in the morning, and commute back home in the evenings if he needs to spend the days there with you both.
Hopefully, you can get something sorted with the hospital from tomorrow, but if they cannot help, then realistically, you either need to pay for accommodation, or swap every few days with each other, or your DP commutes.