Ask a landlord-tenant lawyer and get answers ASAP
The key that you are going to have to point out is that your medical condition is a result of the AC breakdown. If a doctor can verify that the condition that you are having now, the pneumonia, was directly caused by the breakdown, then you can argue that the apartment is no longer habitable. That is the key when it comes to either breaking the lease or taking additional steps, whether the property is habitable. If you do not hear from your landlord quickly, I believe you have a couple options. One you can go to a hotel since that will be air-conditioned and then advise the landlord that you are going to the deduct the cost from your monthly rent. The second option that I see would be contacting a contractor and get an estimate to do the repair. Depending on the cost, you could do that and deduct that from your rent payment as well.