Well of course any unnecessary expense feels like a loss, and if you're not getting the service/product you paid for, of course you'll feel aggrieved. Though in the grand scheme of things two month's extra rent is not a disaster.
In this case I would pay the two months' rent to get out, when moving out be ultra-compliant to avoid any issues getting full deposit returned (make sure apartment is left very clean, furniture and items in place as per inventory, photos taken to prove it, any wear & tear not in inventory photographed, documented, and sent by recorded/registered mail to landlord, all keys etc. returned on time, you keep a log of all contact with landlord, receipts for cleaning items purchased etc.). I would ensure that I state in writing that the 2 months' rent is paid to break the contract, but with an unresolved complaint outstanding.
After getting settled in new place, I would then consider trying to reclaim some of the lost rent from the previous place. Not by asking for all of it back. But, for example, landlord may have found new tenant within 2 months so I would ask for say half of the rent back. By written letter, registered mail, stating that you are seeking a partial reimbursement of 2 months rent paid, you would accept that as final settlement, if refused you will submit a claim in local court (at least in the UK this is a good example of what the Small Claims Court is for. You submit online for a nominal fee, a mediator tries to reach a settlement between both parties, if no agreement, case goes to a local court, a magistrate decides. Worst case is that magistrate refuses your claim and you only lose the nominal fee. In UK, a tenant has a standard right to "quiet enjoyment" of the property, if you can prove apartment was not listed as young/lively/noisy area in original listing, so normal quiet enjoyment would be expected and was not provided, issue was reported repeatedly and not dealt with, you should have a good chance of the claim being upheld if your claim was for a reasonable amount, like half/most of the lost rent plus direct costs but not say all the rent you ever paid for the property. This is the UK situation anyway, I don't know what the tenancy law is like where you are). If the process to submit a claim to a local court is too complicated/requires hiring a lawyer etc., I would just write off the lost rent to experience, as I would not think it worthwhile to spend a lot of my time (plus the stress/inconvenience etc.) working on a claim.
