I had the same issue on my HP LJ P1002, so combined some old experience with new input, and this is somethings that will get it to work.
- Download the last official driver for Windows 10, and start the installer. It will unpack the drivers C:\LJP1100_P1560_P1600_Full_Solution, detect the printer, and finally fail during instalation. At this point, leave the printer connected via USB and turned on.
- Navigate to C:\LJP1100_P1560_P1600_Full_Solution\UTIL\ and run application SIUtility64.exe, and disable the SmartInstall feature on this device. It will simply stop presenting the printer as a USB flash drive with SmartInstall feature, and leave it in a "normal USB printer mode".
- You can now try to turn off and turn printer back on, to verify if the SmartInstall is disabled.
- Go to Windows "Printers & Scanners settings", click "Add device", wait till if fails, and choose Add manually
- Select Add a local printer or network printer with manual settings. Select existing USB port, select Have disk and navigate to the C:\LJP1100_P1560_P1600_Full_Solution to search for driver, select your printer and Install it.
This worked without issues.
Hopefully it will help others as well.
Regards.