How can I help my brother who has had a stroke and is in a nursing home?

Be there for him. Your support & willingness to help are what counts. Aid with activities of personal hygiene are important to self-esteem as well as comfort. As your brother's doctor for specific recommendations.
Support. Recovering from a stroke can be a challenging journey with a gtreat impact on the psyche of the patient. Your best bet is to be there for your brother during his recovery and find joy in small steps. Keeping things in perspective and being an anchor for him during a time where he feels lost, would be of great value to him.