Need to ask a number of questions:
1. What is the AHT software? Its the default Apple hardware diagnostics that you can run to test everything. From researching it, a lot of people have trouble with it not going.
2. To confirm: she has a mid 2009 13" Mac Book Pro, using Mac OS V10.11.6, correct? Yes
3. This is critical: are backups being made to an external device? No
4. Has she ever done any disk cleanup/maintenance/repairs, from a software perspective? One actually can do a good amount of disk cleanup on your own, and there are some excellent products available (both free and commercial) that can help you with those tasks. Has done a full OS refresh install, but will look into that.
5. When did the "crash" occur? And did it happen after the new battery was installed? For the last 2 months, battery just installed last week, same results.
6. Who installed the new battery? And is it a genuine Apple battery for her Mac?