How can we satisfy an old woman who can't fast, and cannot make up what she missed to fast, yet is unhappy? Is there any Dua we can say to ease her suffering?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

If this woman did not fast some days although she was healthy and fit to do so, then she must make up the days she missed and repent to Allaah. However, if she failed to fast because of a sound reason, then she has only to make up the missed days. If she cannot fast, she should give an expiation for each missed day a Mudd (Mudd is equal to the weight of two thirds of a kilogram) of the most common food in her area to a poor person.

As regards now, if she cannot fast, Allaah does not burden a person of what is beyond his scope and ability.  In this case, she should expiate if she is able to do so, otherwise this expiation remains as a debt until she becomes able to pay it.

With regard to calming her down, you should mention to her the Islamic rulings about this issue and that she will not be punished for any act of worship that she failed to perform because of being unable to do so.

Finally, you should mention to her the importance of being contented with Allaah's Predestination and that she will be rewarded for such contentment.

Allaah Knows best.