Oh, Martha,
((((((((((hugs)))))))))))) I'm so sorry your mother had a fall and such an injury. What a worry, not to mention her confusion, pain, and the threat of pneumonia. She's fighting because she cannot understand the pain, immobility, and the unfamiliar surroundings. As you said, at her age it's not good.

There should be a large note on the wall over the head of the bed your mother is in to say she has Alzheimers. She's not being difficult, she's scared and hurting. Sedation is good.
It is important to stabilize the injury ASAP and that does mean surgery. Her injury is likely to be a break at the ball at the top of the femur or an upper thigh break which would be repaired with a metal plate, a rod, or a total hip or similar. Of course, the kind of repair all depends on where the break is. Even with your mom's age and other medical problems, this injury must be fixed and the recovery will be long. She will probably be sent to a rehab hospital when she's recovered enough to leave the hospital. It may difficult for your mom to return to Bill's at all. Just be aware of that and discuss it with Bill while you're there. If my surgery experience can help you, just ask your questions.
As I've looked into train v plane trips, the cost of the train was close to that of the plane. Since you're in the area of many trains on the NYC to Chicago run, perhaps the train will be more economical for you.
I hope that you can get to your mother soon, that the other grandparents can step in to care for the grandsons so you don't have to look at the calendar. Sadly, all of your time and love cannot fix this. You may not be able to stay in NYC for the whole recovery, so let that be okay if that's how it happens.
Take good care of you, too.
Hugs and blessings - Barbara