Chosen Solution
iPhone 5s. The display (and apparently to phone) does not work unless the upper left corner is depressed (pinched between thumb and finger), when depressed it moves noticeably, about 1/32 to 1/16 of an inch. If it is held depressed, display and touch sensitivity seem to work fine. Opened the phone, found a screw that was not in its place, and one that was loose (I tightened it). The loose one was in the area of the upper left in the screen assembly. I could not find where the displaced screw is supposed to be.
Sounds to me like you need a new display. It should work fine, upright or closed. Sounds to me like the cable is the issue. I would try to get that display exchanged from where you got it.
