have 表示“有”的意思时，其宾语通常不与定冠词连用。定语从句所修饰的先行词前面可以用定冠词，也可以用不定冠词 ，例如：
The gentleman whom she encountered addressed her with courtesy. 她相遇的那位先生很有礼地和她讲话。
I have just met a lady (whom) I saw last week . 我刚遇到一位我上星期见过的女士。（《薄冰高级英语语法》p.471）
所以，Mrs. Taylor has ______ 8 years old daughter who has ______ gift for painting—she has won two national prizes. 的答案分别填 an 和 a 是正确的。