Do I have a legitimate case for medical malpractice against my child’s pediatrician?

Medical malpractice is negligence.  Negligence is the failure to exercise due care (that degree of care that a medical practitioner in the community would have exercised under the same or similar circumstances to prevent foreseeable harm).
Prior to filing a lawsuit for negligence against the pediatrician, it may be possible to settle the case with his malpractice insurance carrier. Notify the malpractice insurance carrier in writing of your daughter's claim.   Your daughter's claim filed with that malpractice insurance carrier should include her medical bills and medical reports.  
Compensation for the medical bills is straight reimbursement.  The medical reports document your daughter's treatment and are used to determine compensation for pain and suffering which is an amount in addition to the medical bills.
If the case is settled with the malpractice insurance carrier, NO lawsuit is filed.
If you are dissatisfied with settlement offers from the malpractice insurance carrier, reject the settlement offers and file a lawsuit on behalf of your daughter against the pediatrician for negligence.
You will need to be appointed guardian ad litem to file a lawsuit on behalf of your daughter because she is a minor.
If the case is NOT settled, the lawsuit for negligence against the pediatrician must be filed prior to the expiration of the applicable statute of limitations or your daughter will lose her rights forever in the matter.

We hope you don't have a viable or worthwhile malpractice case, for the reasons we will explain below.
There are two components to any lawsuit, including malpractice: 1) liability, or some wrongdoing (either intentional or through unreasonable carelesness) that makes another party liable, or legally and financially responsible, for injuries, cost, harm, etc.; and 2) damages, or the costs, harm, etc. provably caused by the wrongful act. You need both, because even if someone else is at fault, you can only recover for the provable harm, in an amount proportionate to the harm. 
It is reasonably likely that there is liability: the doctor seemed to have been careless and misdiagnosed your daughter, by ignoring or misinterpreting symptoms.
However, the issue is, what do those 10 days of untreated leukemia mean (other than stress, since there is no compensation for stress in cases like this)? What would have been different had the diagnosis been made 10 days earlier?
If the delay harms your daughter in some significant way, such as reducing her life expectancy, her odds of survival, or quality of life to an appreciable degree, or causes you to incur signficantly higher medical costs, then that is "damages" worth suing over, and you should consult with a medical malpractice attorney (many provide a free initial consultation to evaluate a case; you can inquire into that before making the appointment). 
But if the delay in diagnosis did not hurt your child, you would not have a worthwhile case, since with no harm from the delay, there is no compensation to sue for or to which you'd be entitled. That's why we say that we hope you don't have a case: we hope this delay did not harm your daughter.
If we may, please accept our best wishes for your daughter, and our sympathy for what your family is going through.