Are the Layl (night) prayer and supplication after the Tashahhud acceptable despite forgetting the Taslim, especially since this occurred in the last ten days of Ramadan?

"Salam" (the salutation of peace) is a pillar of prayer according to the majority of scholars (Malikis, Shafi'is, and Hanbalis). Their evidence is the Prophet Muhammad's (peace and blessings be upon him) saying: "The key to prayer is purification (wudu), its sanctification (tahrim) is the takbir (Allahu Akbar), and its dissolution (tahlil) is the taslim (salutation of peace)." As for the Hanafis, they consider the salam to be an obligation (wajib) and not a pillar (rukn). According to them, one may exit the prayer by saying salam or by doing anything else that contradicts the prayer. Your prayer is valid according to the Hanafis, and the acceptance of supplication is an unseen matter entrusted to Allah.