So, a lot of people missed this important point:
I recently bought one of the korean IPS monitors that is hooked to my computer but the monitor only has one input and that is dual link dvi.
The models with a single DL-DVI port have no scaler.
Even with an adapter the xbox will at best
be letterboxed to the exact output resolution. For reference 720p = 1/4th of the screen, which about the size of a 13.3" laptop display on a 27" 1440p. (aka damn small)
At worst it won't work at all. (this is also the most likely outcome)
Good news: You got a very nice IPS 2560x1440 monitor without a scaler which makes it excellent for low input lag PC gaming
Bad news: No KVM or adapter will do what you want.
Solutions: either use another second screen suited for the xbox, or trade/buy/sell/swap/whatever for one of the korean models with a scaler and multiple inputs. (e.g. microcenter carries one)