Learn Finnish: Miten hoidat nämä latvat? - How do you treat these ends?

Yes, you could say Kuinka hoidat nämä latvat? Both words mean 'how', but miten is often preferred when asking about the specific manner or method of doing something. Kuinka is also completely correct here, though it is especially famous for asking about degree or amount, like kuinka paljon (how much).

I thought the verb hoitaa (to care for / treat) required a partitive object. Why is it nämä latvat instead of näitä latvoja?

Hoitaa takes a partitive object when describing continuous, ongoing care (like hoidan koiraa, I am taking care of the dog). However, when it means to handle, deal with, or complete a specific task, it takes a total object. Here, the speaker is asking how you completely deal with this specific set of ends, so the plural total object (nämä latvat) is used.

What exactly does the word latvat refer to here?

The base word latva typically refers to the top of a tree or the very tip of something. In everyday situations, you will most often hear it at the hairdresser: latvat refers to the ends of your hair. This sentence is likely asking someone how they treat their hair ends to prevent damage or split ends.

Why doesn't the pronoun nämä have a matching -t ending like latvat?