Right now the only choice is Arnold Palmer Children's Hospital in Orlando. In October a new children's hospital is opening closer to Disney. I don't know where you are from but if you are used to a level 1 trauma hospital you will be frustrated. I have 2 kids with spina bifida their medical care is done at University of Michigan in the summer months where every specialty known to man is at the children's hospital. We live in Florida in the fall and winter so our emergency care is at Arnold Palmer. Arnold Palmer is part of a medical complex and shares it's specialty doctors. So the neurosurgeon isn't in the hospital but comes from the cancer center in the complex. I took my youngest to the hospital in Feb for a headache and chest pain thinking she may be having shunt failure since she has a VA shunt. The neurosurgeon had them xray her shunt so he could see if the setting was still the same. It was so he ruled out shunt failure. I wasn't very happy that he didn't come over to the hospital to see her but she was admitted for something else and he did come over 2 days later to see her. I just have to keep telling myself U of M is ranked 4 and Arnold Palmer is ranked 24 in best children's hospitals.
All rooms are private but very small and they do provide a roll away bed for a parent to stay. Here are the directions from Disney to the hospital.
Take I-4 East to Exit 81BC at Kaley Street. Turn right on Kaley Street. Then turn left (north) on Orange Avenue. Turn left on Miller Street. Arnold Palmer Hospital for Children is approximately ½ mile ahead on the left.