Q: Software conflict

shut down

disconnect all external devices

boot into safe mode and log into the account

Try safe mode if your Mac doesn't finish starting up - Apple Support

open disk utility (macintosh HD/applications/utilities/disk utility)

select the boot hard drive and click repair permissions (if you misrepresented your last OS "10.10" instead of El Capitan 10.11.x then this will not be available to you in disk utility.

you can also boot into the child account and access your account files by modifying permissions which should not be taken casually and probably not attempted until you figure out what the issue is with your backup. if you would like to specify include the details in this thread.

In any event you can target disk your mac to another computer and access the drive from a separate computer.

Share files between two computers with target disk mode - Apple Support

FYI, this may not be relevant, but the last OSX update would be 10.11.5, not 10.10