R Yirmiyohu Kaganoff has an excellent article covering many points relevant to your question. In summary
- an adopted child can convert, instead of his parents accepting responsibility for the child decision, it is the rabbinical tribunal (beit din) supervising the conversion which acts as the child’s surrogate parents and accepts his conversion
- if the child convert decides on reaching maturity (12 for a girl, 13 for a boy) that he does not want to be Jewish, he invalidates his conversion and reverts to being a gentile (Igrot Moshe YD 1:162)
- once the child achieves maturity and is living an observant lifestyle, this is considered an acceptance of the conversion that cannot be rejected afterwards
- if the child wasn't told before his bar/bat mitsva that he is a convert, and only discovers it as an adult, R Moshe Feinstein writes he would have the option at that time to accept or reject his Judaism, and the parents have limited influence on his decision
As such, if the person you describe has been properly converted (see below) and hasn't renounced Judaism at age 12 or when discovering she was a convert, she is fully Jewish. One does not need to reaffirm one's Jewishness and does not lose one's Jewish status.
Two further important notes
- The question whether it is possible a priori to covert a child in a non-religious family is very involved, see e.g., here for a description of the issues, including the opinion of the Sridei Eish that the beit din has no authority to implement such a conversion
- It is not obvious such a child could be converted today. For instance, in the US, the RCA policies (here, see 5b) require that the child be raised in an observant family and attend Jewish day school for 12 years.
For further reading, see also here.
I assume you are asking in theory. Any practical application of the above would require consultation with a rabbi who is familiar with conversion practices.