Yes, through you, your child is entitled to British citizenship. Unfortunately, this doesn't give the child's non-UK citizen parent any right to a visa or citizenship. Her options are the same as anyone else's: marry, get a job with a qualified sponsor, invest a lot of money in a UK business or start a UK business or enroll in an advanced degree program.
She can't switch from a YMS visa to any points based visa (work, school, investor, entrepreneur) or a visitor visa while in the UK. She would have to return to Japan to apply for any points based visa. She can switch to a spouse visa from within the UK.