Which sentence is better, if I change "in a way that" to "how", should I leave "in"?

Your question is based on practice in constructing relative clauses. One type of relative clause uses a relative adverb. There are three relative adverbs:

• when: of time (Do you remember the time when I broke my arm?)
• where: of place (London is the city where I was born.)
• why: of reason (I'd like to know the reason why you said that.)

(More information here: https://www.grammarwiz.com/relative-adverbs.html)

You cannot use how as a relative adverb. So none of the following sentences is grammatical:

• *Termite mounds are built in a way how hot air rises out and cool air comes in.
• *Termite mounds are built how hot air rises out and cool air comes in.
• *Termite mounds are built in how hot air rises out and cool air comes in.

The original sentence is fine, although I think it can be slightly improved by the addition of such:

• Termite mounds are built in such a way that hot air rises out and cool air comes in.

For me, this gives a little more emphasis to the intention behind the building method.