If a woman did not forget, but was in an emergency situation where she could not light. For example, attending to a parent as he/she is being admitted to the hospital for a life treatening event (ie. cardiac event). When things were settled it was after shkia. Would she be expected to light an extra candle indefinitely?

Shalom, Thank you for your question. If a woman did not light Shabbat candles, not because she forgot, but through no fault of her own (such as in the situation you wrote about), then she does not have to add an extra candle to her lighting in the future (See Shmirat Shabbat KeHilchatah, 43,5). Blessings.