A similar entry appears in the Wheel of Time Companion confirming the information available in the main story arc.
Author: Atarah al'Norahn
Innloine is one of Moiraine Damodred's older sisters, making her a high-ranking Cairhienin noble. She has one other sister, Anvaere. She has children, so it is likely that she is married and possible that she no longer bears the Damodred name.
According to Moiraine, Innloine is a warm and loving mother. She is stubborn, but she is not terribly bright.
When Jarna Malari asks Moiraine if either of her sisters have a chance of claiming the Sun Throne after Laman's death, Moiraine replies her sisters are well thought of, but not for the throne. Everyone knows that if Innloine won the throne, affairs of state would come a poor second to playing with her children, and that she would likely forget about the affairs of state in caring for her children. Moiraine also thinks to herself that Innloine's stubbornness and lack of brightness would be a dangerous combination in a ruler. Moiraine says that no one will support Innloine for the throne, even within House Damodred.
(Reference: New Spring, Chapter 6)