For the emitters, one cheap LASER-pointer and one tiny mirror could be used, with some thin polycarbonate (or acrylic) plastic in small slices used as transparent mirrors.
That way you'd have exact control over where the beams hit - and save all the LEDs
Whatever method, always place the horizontal row of emitters at the bottom. That way the receivers won't see as much of the ambient light.
Did you consider buying one of the loose IR-frames to mount on a monitor for the very same purpose? (They were all the rage back in time).
And there are several other ways of getting touch on a screen, but if it hasn't got at least half the resolution of the monitor (so that you can draw directly on the screen, I fear that you'll get very bored with it. Either use a keyboard, a mouse or touch - don't make yourself a situation where your hands have to scoot all over the place all the time.