The woman said that she had the implant because she and her husband already had two children and did not want more, and was told that it would prevent her getting pregnant for three years.
The judge said the private hospital in Nairobi had been negligent.
The Aga Khan University Hospital has not made any comment.
Despite being served the court papers, no representative from the hospital appeared in court, the judge said.
The woman told the court that she became pregnant a year after the procedure.
She said the unplanned pregnancy caused her emotional pain, financial strain and marital problems, partly because her husband thought she had lied about seeking contraception.
The High Court ordered the hospital to pay the mother $5,000 in damages and $43,000 towards the cost of bringing up the child.
“Doing the best that I can with the material before the court I will award 20,000 shillings [$203, £132] per month for 18 years. In arriving at this figure, I have taken into account that the cost of living will probably keep rising as the years go by,” Kenya’s Standard newspaper quotes the judge as saying.
The implant is more than 99% effective as a form of birth control when used correctly, according to the UK’s National Health Service.BBC AFRICA