NVidia driver not loaded

Question

I have installed Ubuntu on my PC a few weeks ago because I bought an SSD and wasn't able to reinstall Windows on it.
Recently, I tried installing Nvidia graphics drivers (for my GTX765M) so I could play games. I first installed X.org/nouveau but it changed my laptop's builtin screen resolution from 1920x1080 to 960x540 (my external monitor was still in its normal resolution though), and it made my games even slower than when I used the integrated GPU (about 4 FPS), so I gave up on this driver.

I then used APT with Ubuntu's graphics-driver PPA to install actual Nvidia drivers (especially nvidia-driver-396). After I spent a whole day fixing some corrupted APT/Deb packages that had been downloaded, I finally managed to make the driver appear in Ubuntu's Software & Updates / Additional Drivers list, and it was checked. I rebooted, just to find out that it didn't work, and I still had the resolution problem.

To diagnose the problem, I then typed sudo nvidia-settings in my term, the settings window opened as normal, but it logged:

ERROR: NVIDIA driver is not loaded

ERROR: Error querying enabled displays on GPU 0 (Missing Extension). 

ERROR: Error querying connected displays on GPU 0 (Missing Extension).

I assume that because the driver is unloaded / failed to load, Ubuntu used Xorg as a fallback, hence the wrong screen resolution, but I have no idea of how I'm supposed to start the Nvidia driver. Is it a service ? A kernel module ? Currently I can only use my laptop as a desktop computer in one-monitor mode on my external screen, which is obviously annoying.