Only HDMI 2 officially supports 144hz at 1080p. 1.4b officially supports 120@1080, but only for what they call "3D processing", which I'm not sure what they even mean by, maybe Setsul can explain (I think it means it just copypastes the same 60hz signal to output it twice for 3D viewing), but it's not 2D viewing aka what you want.
Anecdotal evidences exist of people hacking and OCing their way to 120hz over HDMI 1.4 but basically, it's not supported.
You should get a new DL-DVI cable and hope for the best and get your monitor fixed if the input is really fucked.