What vinegar acidity is necessary to effectively kill household mold

IMO you are making several leaps here that don't fit. You've got mold below a window, spread out about 2 feet on either side, and you suspect condensation as the culprit.

You specify that the damage is old - years old by the look of it. Condensation happens every year, every season. Why would the damage be old, it should be fresh if it's condensation.

Finally - if condensation were the culprit, the condensation and thus the mold would be uniform. Water leakage through a window, however, is going to fan out below the window in both directions...

The answer is your window leaked at some point - years ago by the sound of it since the damage is old.

Solution?

Remove and replace ONLY the section of drywall that is damaged. Do not add a partial vapor barrier as that's about as useful as putting in an aquarium with only 2 sides. Mud and tape properly and repaint.

What you describe requires major plumbing work and can cause devastating damage to your home. Your landlord should be responsible for fixing this because with moisture leaking behind the walls like this then it can quickly create major property damage to his rental unit and unhealthy mold.

Your landlords refusal to fix this on his own is creating a health hazard for you, and in many countries and states there are laws that protect tenants and give them certain rights in a landlord-tenant relationship. One of these rights is the right to a safe and hazard free home, which means you can and should be able to bring suit against the landlord in the event where he refuses to fix a situation with the property that is creating a signficant health hazard for you and your family.

To get rid of the mildew, talk to your landlord and tell him to fix the leak immediately.