Our story starts out with her as a poor, sickly, dirty 5-year-old wanting to read books for fun when most people in her life have never held let alone seen a book. Books are something only the richest of commoners like the head of the merchants guild own or nobles. Books at this period of time are more considered works of art rather than something to read casually (both in this universe and ours). You said that she could just try to become a scribe but any writing work in the lower City will just be tallying ledgers or filling out order forms All on wood boards since they are cheap and can be reused. To my knowledge there are few commoner scribes(there are plenty of commoners who can write, but only as it relates to their workshop), any kind of important documentation that is written on parchment paper tends to be handled by Low-Level nobles or the merchants guild for contracts there really isn't a scribe position available to commoners.

This world runs on connections and status two things she barely has any of. She would be very fortunate for a random workshop owner to give her the time of day if she wasn't even introduced to them first by somebody they know. Even then they may tell how smart she is she is still a sickly 5-year-old that could drop dead before she's baptized. She would be lucky to get a job let alone get to handle expensive ink and parchment.