Angels have the Phone Box
Thanks for the file.
"MicrosoftEdgeSquare44x44.scale-125_contrast-white.png" - hash mismatch (nothing to worry about)
"adcjavas.inc" of Microsoft-Windows-Microsoft-Data-Access-Components - hash mismatch
You've turned off Fast Startup, correct?
Please also uncheck "Give me updates for other Microsoft products when I update"
Settings>Update & Security>Windows Update> Advanced Options
Shut the computer down. Since the fan and power button will still be on, hold the power button down until the laptop completely shuts down. Remove the battery and just use the power cord. Turn the computer on and let it boot up completely. Shut it down again. Turn it on again. Try Windows update. The most recent cumulative update that's come through is KB3081438.
If it's still failing, error 0x800F081F is usually resolved with a clean install. Or, you could try an in-place upgrade, (repair install), to keep all your files and apps.
Before you begin:
Make sure your data is backed up.
Verify Windows is activated. (Settings>Update & Security>Activation)
Remove any 3rd-party antivirus and disk monitoring programs.
Have your program disks and keys available if doing a clean install.
Disconnect all peripherals (external drives, printers, etc.)
Let us know how it goes.