What a tough situation.
Keep in mind that there is a reason that they can't tell you the schedule today. Surgery schedules change based on the urgency of cases. They can only plan so much in advance, and if a more urgent case than your daughter's comes up tomorrow, they will schedule her ahead of your daughter.
I'd ask your child care provider to be flexible with the schedule for Friday. Let he/she know that you can't get a firm schedule from the hospital until tomorrow afternoon, and ask her to be open for the day so you have it covered.
Can she have anything by mouth on Friday--clear liquids, or nothing at all?
ETA: Just thought I'd address the idea that the healthcare providers are greedy and lack compassion. As I type this I am waiting for my husband to text me to tell me he's done seeing his last patient of the day. It is 9 pm. He left the house this morning at 6:30 am. He has been taking care of patients for 14 hours so far today. He was also on call last night, and it was a particularly busy call night, and we were both awoken at least 8 times during the night with pages from the ER. Does he do this because he's greedy? I'll bet someone thinks so, but I know that is not the case. Why does my husband work so hard? His single largest driving factor is his concern for his patients. He knows that there is already a defecit of physicians, and that in a few years, when 32 million newly insured patients enter the market, there will be an even more severe shortage. My kids haven't seen their dad since they went to sleep on Monday night. They won't see him again until Saturday morning (his first day off in weeks), unless they get up before 6:30 am to catch him as he heads out the door. This is a very typical week for us.
In case you need some encouragement that those who are caring for your daughter on Friday MIGHT just be good people, I thought I'd share with you this story about a pediatric surgeon who died this weekend while rescuing 2 children from drowning in Lake Michigan. It might help give you a bit of hope and faith in your daughter's care providers. http://www.chicagotribune.com/news/local/ct-met-doctor-dr...
ETA2: So happy to hear that the day of the surgery went smoothly and didn't cause anyone undue hardship. Best wishes to your daughter for a speedy recovery!