Ascoli Piceno, as you know, is a province in Central Eastern Italy. It's slightly southeast of Siena but boy what a train journey to get there. You can get there by Trenitalia train. The trip takes 7.0-8.0hrs with 2-3 train changes. The fastest runs take you primarily via Florence, Bologna, Rimini, Ancona and down the eastern coast. The slowest run, which there are fewer, routes you south almost to Rome before going east to Foligno and then to Ancona on the east coast. CLICK HERE to see the train schedule.
Since your daughter is spending the summer there, I might suggest taking a 3hr train ride to Cortona and spend a night or two either there or Assisi (see both towns). (The southern train route goes through Foligno which is 20 minutes from Assisi). These are fabulous towns. Then, proceed to Ascoli Piceno. This can be a 4hr-45min train trip if you get the right train run. CLICK HERE to see these runs.