Is she obliged to fast even though her body is small and she does not menstruate, and she is fifteen years old?

Fasting is obligatory for every adult Muslim of sound mind who is able to fast.

Adulthood is reached when a person completes fifteen years (in terms of age, based on the Ilsamic lunar calendar), or emits maniy whether awake or asleep, or when coarse hair appears around the private part. In the case of women, the onset of menstruation is also a sign of reaching adulthood.

It says in al-Mawsu‘ah al-Fiqhiyyah (8/191):

Reaching adulthood on the basis of age: this applies when none of the other signs of reaching adulthood have appeared before that.

The fuqaha differed concerning the age of reaching adulthood.

The Shafi‘is and Hanbalis, and Abu Yusuf and Muhammad ibn al-Hanafiyyah, thought that a person, male or female, is to be regarded as an adult upon completing fifteen lunar [Hijri] years. The Shafi‘is stated clearly that this is the correct age, because of the report of Ibn ‘Umar: I was shown to the Prophet (blessings and peace of Allah be upon him) on the day of Uhud, when I was fourteen years old, and he did not allow me [to fight], as he did not think that I had reached adulthood. I was shown to him on the day of al-Khandaq (the Trench) when I was fifteen years old (i.e., completed 15 years), and he allowed me [to fight], as he determined that I had reached adulthood. [Narrated by al-Bukhari].

Ash-Shafi‘i said: The Prophet (blessings and peace of Allah be upon him) sent back seventeen of the Sahabah when they were fourteen years old, because he did not think that they had reached adulthood. Then they were shown to him when they were fifteen years old (completed 15 years), and he permitted them [to fight]. They included Zayd ibn Thabit, Rafi‘ ibn Khadij, and Ibn ‘Umar.

The Malikis thought that adulthood is reached at the age of eighteen years, or when a person passes the age of seventeen and begins his eighteenth year. Al-Hattab noted five views in the madhhab. According to one report it is eighteen years, or it was said that it is seventeen years. Some commentators on ar-Risalah mentioned also sixteen years, or nineteen years. And it was narrated from Ibn Wahb that it is fifteen years, because of the hadith of Ibn ‘Umar quoted above.

Abu Hanifah thought that reaching adulthood on the basis of age occurs at eighteen years for a boy and seventeen years for a girl, because of the words of Allah, may He be exalted (interpretation of the meaning): {And do not approach the orphan's property except in a way that is best until he reaches maturity} [al-An‘am 6:152]. Ibn ‘Abbas (may Allah be pleased with him) said: Adulthood is reached at eighteen years of age. This applies to boys; girls reach adulthood earlier, so for them it occurs at seventeen years of age.

The majority are of the view that adulthood on the basis of age is reached when a person completes fifteen years.

Based on that, if this girl has not yet begun to menstruate or emit maniy, and no coarse hair has grown around her private part, then when shecompletes fifteen years she is regarded as an adult and is obliged to fast. Body size makes no difference.

As that is the case, she must check whether the other signs of adulthood have appeared, in order to determine whether she is obliged to fast or not.

But if she is not able to fast, or if fasting will harm her, then she must ask the doctors whether there is any hope that she may recover from her sickness in the future so that she will be able to fast, or is there no hope of that?

If there is the hope that she will recover, then she may not fast, but she must make up the missed fasts when she is able to do so. If there is no hope of her recovery, then she may not fast, and she must feed one poor person for each day, but she does not have to make up the missed fasts.

And Allah knows best.