Gentimycin kills the labyrinth. They have to be careful to only administer it to the balance organ. If they get it in the hearing section, it will kill that too. With your balance organ killed off, there will no longer be any bad data transmitted by it to cause dizziness. It should take a period of time for your brain to get used to no longer having a balance signal from that side, and then, all should be well in terms of dizziness.
Your balance function from your AN side would eventually go away anyway. This injection just gets it all over with in one shot so as to make all of the dizziness and adjustment happen in one incident rather than randomly and over a period of months or years. It's like pulling a bandage off quickly rather than slowly.
These injections are sometimes used for Menier's disease when other methods don't work.