The law does recognize an action for wrongful birth. Parents of a healthy normal child born after an unsuccessful sterilization operation may not recover the child-rearing costs against the physician, but may recover the expense, pain and suffering, and loss of consortium associated with the failed sterilization, pregnancy, and childbirth, since damages may be established with reasonable certainty, and do not invite disparagement of the child involved. However, the parents of an unplanned, but healthy, child may not recover child-rearing costs from the physician who performed the unsuccessful sterilization procedure.
You may be able to sue, however, you can only be successful if you can prove the physician fell below the applicable standard of care. The standard of care is the minimal level of competency in the community at that time. If the surgeon was a Board Certified Urologist, it is a national standard of care. Also, you need to show you did not know and should not have known of the substandard care 10 years ago or else the statute of limitations may have passed.
Vasectomies are known to fail even if the surgery is done correctly so the fact it failed is not enough to show fault.
To win, one of the things you likely must prove is that the child is actually yours biologically.
The human body is sometimes quite amazing. Damaged body parts sometimes repair themselves. It is not unknown that tubes that are cut or tied become functional again.
The vasectomy procedure apparently was effective for about 10 years. You may run into statutes of limitation problems.
I would encourage you to investigate the matter; however, the human body is amazing and sometimes a properly performed vasectomy or sterilization will fail due to natural processes such as recanalization and fistula formation. The fact 10 years passed without incident would make me suspicious that the procedure was done properly and the late failure was due to the resiliency of the human body. Also there may be statute of limitations issues given the time that has passed. That would be a question under the law of the applicable jurisdiction.