Is there any basis for the belief that reciting Surat Al-Layl and Surat Ash-Shams seven times for seven nights, and then supplicating with "O Allah, grant me relief and a way out...", helps in fulfilling needs?

The entire Quran is a blessing, a healing, and a mercy. There is no specific evidence that singles out two particular surahs, with a specific number of recitations, for fulfilling needs. Worship is based on divine guidance (tawqif), meaning Allah is worshipped only through what He has legislated. The Prophet, peace and blessings be upon him, said: "Whoever introduces into this matter of ours that which is not from it, will have it rejected." Therefore, one must avoid adhering to a specific number of repetitions in remembrances (dhikr) and recitations for which there is no evidence.