Is it obligatory for a sick woman to seek the help of her father or brother for istinja' (washing after defecation or urination) and ghusl (ritual bath) if no other woman is available to assist, given her feeling of embarrassment and discomfort?

The default is that it is forbidden for men and women to look at or touch the 'awrah (private parts) of others, except between spouses, and for those in necessity. This is based on the hadith (definition): "A man must not look at the 'awrah of another man, and a woman must not look at the 'awrah of another woman." It is permissible for a sick person to combine two prayers. It is permissible for a father or brother to clean a sick female patient if her mother is not present and a woman cannot be hired. The cleaning must be done without looking or touching, if possible. If looking is necessary, it is permissible due to necessity, as necessities permit prohibitions, and they are estimated according to their measure.