Contrary to popular belief, it's nearly impossible to sue a doctor and basically impossible to sue a hospital. Hospitals have a huge sum of money to spend on lawyers and legal defenses. The only way you can sue a hospital is if many people have been injured and you can get class action status.
Get a medical malpractice attorney or a hospital errors lawyer.
But before you randomly pick a name off the yellowpages, consider the following: the experience of the lawyer (check their websites if they really handle medical malpractice and if they've won cases) and it would be a bonus for you especially since you're going up against a big hospital to get a lawyer who handles cases on contingency fee.
It's more beneficial for you since you don't really have to pay them until you win the case, in short, "No Win, No Fee".