Are my prayers that I have performed for 13 years without saying "wa rahmatullahi wa barakatuhu wa 'alaina" (and the mercy of Allah and His blessings and upon us) in the final tashahhud valid or void?

The sufficient form of the tashahhud — according to those who deem it obligatory — is: "All greetings [and salutations] are for Allah; peace be upon you, O Prophet, and the mercy of Allah; peace be upon us and upon the righteous servants of Allah. I bear witness that there is no god but Allah, and that Muhammad is the Messenger of Allah." In the final tashahhud, "Allahumma salli 'ala Muhammad (O Allah, send blessings upon Muhammad)" is added after it.

The tashahhud is not obligatory according to the Hanafis and Malikis, whereas it is an obligatory pillar of the prayer according to the Shafi'is and Hanbalis. The preponderant opinion is its obligation, but it is permissible to adopt the less preponderant opinion when the action has already occurred and it is difficult to rectify it. Therefore, there is no harm in not repeating past prayers, provided that one adheres to the obligatory tashahhud in the future.