Learn Polish: Małe dziecko dobrze się z nią bawi. - The small child plays well with her.

After the preposition z (meaning "with"), Polish requires you to use the Instrumental case. The base pronoun is ona (she), and its Instrumental form is nią.

I learned that "with me" is ze mną. Why is it z nią and not ze nią?

The preposition z only changes to ze to make pronunciation easier when the next word starts with a difficult cluster of consonants (like the mn- in mną). Because nią starts with a single consonant sound, the standard z is easy to say, so it doesn't need the extra vowel.

Why is the word się here? The child isn't playing with "itself".

In Polish, the verb for "to play" (in the sense of kids having fun) is bawić się. The się is a permanent, mandatory part of this verb. If you drop the się, the verb bawić changes meaning to "to entertain" someone else.

Can I put się at the end of the sentence next to the verb?

It is best not to. The word się is a "weak" word that hates being at the very end or the very beginning of a sentence. Placing it earlier in the sentence, like after dobrze, sounds much more natural in Polish.