I have done LED in kitchens. The usual setup is to install a receptacle inside the upper protion of a cabinet and plug the transformer into it. All feed wires feed from that point.
In your case, I would tap into the receptacle across the hall and install a receptacle high up, to the hinge side of the closet. Since the feed wires (you need a power leg and a switch leg) are low volt, you can drop them down from there along the hinge side of the door jamb inside the wall (you are OK with some drywall damage, yes?). Install the switch in the hinge side by drilling through the framing (jamb + jack stud + kind stud). Run the feed wires to the switch, connect them, install the switch.
Now you need to run the switch leg to your first LED light strip, then any others, sequentially.
An alternative would be tu use a motion sensor instead of an embedded switch. I have done this in walk-in closets. If you REALLY want to have a WOW factor, try this ... Run the power from your source receptacle to the closet ceiling. Install a switch box with a motion sensor. Run the switch leg over to another receptacle mounted high and behind the wall with the door. The receptacle will be activated when the motion sensor turns it on. Plug your transformer into this receptacle.
Now, you can run your LED hookup wires inside the wall for a clean installation. The picture shows this arrangement on the sidewall ... you could also do it this way if you like.