When you restored to factory settings, I'm assuming you also put the hard drive back to factory settings.
So, the first thing to do is to get the latest video drivers. That might allow options that the earlier drivers did not.
If your laptop is 1366x768, that's not a good HD resolution which the Sony might want. So, the next thing I would try is to setup for dual screens (laptop screen + external monitor) and then setup for non-clone or extended screen mode. This way you can set a different resolution for the external monitor. Set that to 1920x1080.
Usually the video driver software comes with a set of display suite software that provides an easy way to do this. A right click on an empty spot on the desktop should give you option to bring that up. Intel Graphics control panel is the one for your laptop. Use that to look for dual screen mode.