I once had problems getting an OpenGL program running under Windows on a 2nd gen i5 laptop, and discovered that the issue was that Intel had actually removed support for anything later than OpenGL 1.1 from the later versions of their drivers for those iGPUs.
After a little research, it looks like there’s a “legacy” version of Mesa (mesa-amber) recommended for iGPUs on anything older than 3rd gen, and also that even the main Mesa drivers don’t support Vulkan fully on 3rd and 4th gen Core iX processor iGPUs.
I think that you may just be out of luck trying to use Vulkan on a 2nd gen Core iX CPU iGPU, but I’d love to hear from someone who knows better.