"Le lee un diario a ella" is a phrase with an implicit subject. I think this is a figure that doesn't exist in english, where every phrase has to have it's subject.
In spanish, however, it's possible to avoid writting the subject in phrases where the subject is obvious and well-known.
In this case the subject should be "Él" or "Ella", "Él le lee un diario a ella/Ella le lee un diario" (note that if we use "Ella" as subject we cannot use it as a complement).
We don't need to use an explicit subject when both the speaker and the listener know who is the subject.
Something similar happens with the use of pronouns.
If we were telling this phrase to someone out of context we should use something similar to: "Pedro le lee un diario a María"
When the listener is in the context, he knows that Pedro is reading and María is listening, so the speaker doesn't need to use their names, he can use pronouns for both, and even totally avoid them: "Le lee un libro" will be a perfectly understandable phrase for someone aware of the context where it's said.