Hi rayt2,
Thank you for using the Apple Support Communities! We can hear how important it is to get your sister's iPhone back up and running as normal. Let's work together on a resolution.
It sounds like you've taken great steps already in an attempt to resolve the issue. I'd like you to review the specific steps given in this article, under the section titled, 'If your device turns on but gets stuck during start up', to confirm you tried this method to reinstall the iOS on her device via your computer: If your iPhone, iPad, or iPod touch won't turn on or is frozen - Apple Support.
If you did use the steps above to place the device into recovery mode to reinstall iOS, but it still failed to error 4013, try these error-specific steps next: If you see error 9, 4005, 4013, or 4014 when you restore your iOS device - Apple Support.
Let us know how this works out. Regards.