Your current plan is no good -- first off, 400.8 point 1 forbids the use of cords as a replacement for permanent wiring (stuffing a cord down a conduit certainly counts, and is also prohibited explicitly by 400.8 point 6). Second, 400.9 prohibits the splicing of cord during installation. Third, wire splices need to be in a junction box so that they can be serviced in the future.
A better plan would be to bring the conduit out to the receptacle box where you plan to tap power, and then stuffing NM through the conduit (yes, this is OK). At the box end, you'd simply tie it into the rest of the wiring as you would ordinarily (using pigtails and wire nuts if they aren't already present there). At the monitor end, you'd attach a field-fittable IEC C13 connector directly to the NM cable and plug it into the monitor (you may need a right angle connector here -- make sure it accepts 14AWG as not all do).
Your other option is to use a recessed receptacle in a junction box mounted at the monitor end of the conduit, and plug the monitor's cord into it. This requires that the cord either be exposed completely, or the monitor be readily removable (i.e. "take a few fasteners off, remove a trim piece, and it comes out", not "oh, we have to tear into the wall to replace this") in order to apply 400.7(A) point 8. It also requires a way to get the cord to the recessed receptacle, considering you can't bury the junction box + receptacle in the wall.
Given that you are dealing with an AC adapter -- the recessed receptacle approach is the superior one (I don't believe you can get the smaller IEC connectors to fit on 14AWG anyway!). The main issue will be finding a pairing of recessed receptacle and shallow box that will fit together while accepting the PVC conduit. I'd start with a P&S P108W as that box is only 1.125" deep, leaving 0.375" for the cable and faceplate, and see what recessed receptacles work with it -- the silly folks at Leviton don't publish a dimensional drawing for the 689-W linked above, so I can't tell from here if it fits or not. (Drilling a 1/2" conduit KO in the above box should be OK btw -- the FS-type boxes that are designed to fit conduit are all too deep for this job.)
Oh, and if you do use the recessed receptacle approach, you'll probably need to finish out the monitor-alcove with some half-inch drywall back-to-back with that plywood backplate, using screws and standoffs to mount the electrical box. (Boxes that stick out from their surroundings are kosher by Code.) Of course, the only way any of this will work is if the cavity in the monitor will fit all of the cabling as well as the transformer and the J-box.
Best Answer
Sounds like an issue I recently faced. My wife was drying her hair in our bathroom. Two of my children were taking a shower (running the lights and fans). Circuit breaker popped and that half of the house went dark. Turns out someone had spliced into the light circuit inside the attic (the attic lights made it easy) to add these extra outlets to the bathroom. My wife, who had the misfortune of picking one of these outlets (she has an original outlet on a different circuit), put the circuit over amps.
My guess is your lonely outlet has some friends, and may be a "We need an outlet here" hack into a smaller circuit. Determine what is on this circuit first. If you're sharing your circuit with, say, a refrigerator, that could be a big problem.
One way to potentially solve this is, if it's an interior wall adjoining another room, to see if there's an outlet on the other side of the wall (on a different circuit, or you'll just exacerbate your single outlet problem). If you have this type of setup, measure to where the outlet will be along the living room side (make sure you figure out where the studs are as well). Then you can use an old work drywall box and fish a wire through the hole into the existing box and splice off of it. I've done this myself and it's a great way to add more outlets without having to do any extensive wire work.