While we do not know the exact rooms the Duchess of Cambridge occupied while giving birth, we do know the charges the hospital publishes (.pdf) for its maternity services. One newspaper has estimated that a 2 night stay in luxury rooms would be about £12,000 or US$ 18,000. Gene Declercq, PhD, Professor of Maternal and Child Health at the Boston University School of Public Health calculates the average cost of a vaginal delivery in the USA is $18,329, although in many cases this is offset by insurance.
Once you strip out the extra costs - two senior consultant obstetricians and the luxury "hotel" accommodation that include meals and overnight stays for the spouse plus the prenatal checks that are included in the package price at the Lindo Wing - Americans pay more than royalty to give birth. Remember the Cambridges are paying the real costs of the stay. The Lindo Wing (.pdf of their brochure) where Kate gave birth is part of a NHS hospital but is run as a commercial unit for "private patients" from the UK and all over the world. The object is to add to the income of the NHS Trust that runs the hospital and others in the area.
Gene Declerque's figures are of course averaged and do not reflect the reputation of both the hospital and OB staff so a comparison you would have to, if you can, get prices from somewhere like the Mayo Clinic. He calculates that even after insurance, Americans pay about $3400 out of pocket. The cost of a birth to the NHS depends on where it happens but in a hospital unit (2012 figures) the average is £1,631 or US$2450.
In other words Americans spend more in co-pays than it costs Britain's national health service to deliver free to the parents.
The Republicans of course would have you believe that as the USA has the "best healthcare in the world" these extra costs are worthwhile but Declerque points out:
What's ironic is we can't even claim that the extra expense pays off in healthier mothers and babies. According to a study by the Institute of Medicine, the U.S. ranks at or near the bottom on virtually all maternity care outcomes.(Please note the links to the St Mary's Hospital/Imperial NHS Trust/Lindo Wing may not work as there appears to be very heavy demand on their site.)