Where should I enter into a state of Ihram for a second Umrah with my mother, who is coming from Egypt? Should it be from Jeddah or from Al-Taneem?

Whoever intends to go to Mecca for Hajj or Umrah must enter into a state of ihram from the miqat through which they pass or which they parallel. It is not permissible for them to bypass it without ihram. As for those who are within the miqat, their miqat is from where they initiate the ihram. The second Umrah from Jeddah, its ihram will be from Jeddah. Performing another Umrah on the same day or shortly after the first Umrah is not from the Sunnah.