Learn Italian: Ho le lettere del mittente, ma non gliele consegno. - I have the sender's letters, but I do not deliver them to him.

gliele is a combined pronoun. It fuses the indirect pronoun gli (to him) with the direct pronoun le (them, referring to the feminine plural le lettere). When these pronouns combine, an -e- is added in the middle to make the pronunciation flow, and they are written as one single word.

The translation says "to him". Could gliele also mean "to her"?

Yes, absolutely! When forming combined pronouns, the indirect pronouns for "to him" (gli) and "to her" (le) both transform into glie-. It can even mean "to them". You have to rely on the context of the conversation to know whether you are refusing to deliver the letters to a man, a woman, or a group of people.

Why is the order non gliele consegno instead of putting non after the pronouns?

In Italian, object pronouns generally go right in front of the conjugated verb (here, consegno). When the sentence is negative, the word non must go directly before that pronoun block. The standard order is always: non + pronoun(s) + verb.

Why is it del mittente instead of just di mittente?