To be truthful, you did well not to sign a new lease. By not signing one you automatically became a tenant of a Part 4 tenancy and have all the rights associated with it One of those rights is to be able to vacate without having to find a replacement tenant; you only have to give the correct amount of notice. Therefore, you have not broken the terms of the lease. I believe that the PRTB do accept emails as being in writing (but not where the law stipulates that there must be a signature).
You are liable for rent until the expiry of your notice unless you agreed a shorter period or until new tenants moved in, which ever is the sooner.
If the landlord does not live in Ireland, then the agents may have issues contacting him and he may have to organize the transfer of the deposit to the agents. Ultimately, the landlord is responsible for the return of your deposit. The law requires that the deposit should be returned "promptly" which is decidedly open to interpretation but is generally considered to be a maximum of 14 days. The fact that the agents have had the 56 day notice period to contact the landlord does not show that they are on the ball. They may, in fact be waiting to re-let the property and use the deposit of the new tenant to repay you yours: and so the cycle continues.
You could advise the agents that unless your deposit is returned in full, as verbally agreed, within the next 7 days, you will be making a claim with the PRTB which is a very straightforward thing to do. This might just get them on the ball.