Learn Hardcore Polish: Ta lekcja polega na pisaniu zdań. - This lesson is about writing sentences.

Because lekcja is a feminine noun in the nominative singular, so the demonstrative pronoun must agree in gender and number. Feminine singular uses ta, not ten (which is masculine).

What does polegać na mean, and which case does it require?

Polegać na means “to consist in,” “to be based on,” or “to be about.” It always takes the preposition na, which in turn requires the locative case of the noun or gerund that follows.

Why is pisaniu in the locative case instead of using the infinitive pisać?

After polegać na, you need a noun in the locative. In Polish, the action “writing” can be turned into a noun pisanie (a gerund). Its locative form is pisaniu. You cannot put the infinitive pisać directly after polegać na.

Why is zdań in the genitive plural and not in the accusative?

When you have a nominalized verb like pisanie, any object it “takes” normally appears in the genitive (partitive genitive), not the accusative. Here zdanie (“sentence”) becomes zdań in the genitive plural.

Can I rephrase the sentence using an infinitive or another structure?

Yes. One common alternative is:
Ta lekcja polega na tym, aby pisać zdania.
You could also say:
Ta lekcja ma na celu pisanie zdań.
or simply:
W tej lekcji piszemy zdania.

What part of speech is pisaniu, and how do I recognize it?

Pisaniu is the locative singular form of the noun pisanie (a gerund formed from the verb pisać). You can recognize it because it ends in -u (a typical locative ending for neuter nouns ending in -ie).

How does polegać na differ from składać się z?

Polegać na (“to consist in/be about”) describes the content or essential nature of something.
Example: Ta lekcja polega na pisaniu zdań.
Składać się z (“to consist of/be made up of”) lists the components or parts of something.
Example: Ten kurs składa się z dwunastu lekcji.