Is it permissible for my friend to ask for a divorce from her psychologically ill husband, and would she be sinful if she did so, and can the judiciary divorce her from him without awaiting his consent?

The wife should be patient, seek reward from Allah, and raise the children. If the husband's psychological illness was insanity before the marriage contract and the wife was unaware of it, then it is a defect that grants her the option to annul the marriage. If she became aware of it after the contract and accepted it, her option is forfeited. If the insanity is supervening (i.e., occurred after the contract), it is not considered a defect unless there is a fear of harm to the wife or children. In this case, she should refer her matter to the Sharia court.