Spelling correction: bronchitis, not bronchities
"Sleeping under an air conditioner can lead to respiratory problems, including bronchitis *or/and* even pneumonia."
In this case, I'd say "which includes" doesn't work because "which" is a relative pronoun that signals a non-restrictive clause, and in this case, the two illnesses were important to specify.
"Which includes" is used in terms of clarifying things previously mentioned. In this case, you're not really clarifying, but providing examples.