How can my cousin, who has severe burns and is wrapped in bandages, perform ablution (wudu') and pray during her long period of treatment?

If your cousin is able to wash her face with water without harm, she should do so, then perform tayammum for the rest [of her body]. Otherwise, tayammum is sufficient if most of her body is injured.

If she cannot use her hands for washing or tayammum, she should seek someone to do it for her.

As for prayer, she should pray standing if she is able. Otherwise, she should pray sitting. Otherwise, she should pray on her side. Otherwise, she should pray lying down. Prayer is not waived for her as long as she is conscious.