Add copper wire led lights to existing 12V Landscape Light Transformer

Electronic transformers designed for Halogen lamps will kill LEDs.

Electronic transformers are very simple switched mode power supplies. They "chop" the input 230 V/125 V sine wave in a way that will yield an average output voltage of 12 V. However, the peak voltage will be much higher and will eventually kill LEDs (that's no problem with Halogen lamps because they react more slowly). In addition, the chopped waveform may cause radio interference if the cables to the lamps are too long. The waveform may also be the reason why your voltage readings are off.

Instead of an electronic transformer designed for Halogen lamps, you should either use:

• a conventional AC transformer (the heavy kind); or
• an electronic DC transformer designed for LEDs.

Regarding your code question: no, you cannot hardwire the transformer.

I believe this falls under NEC 400.7:

400.7 Uses Permitted.

(A) Uses. Flexible cords and cables shall be used only for the following:

1. Pendants
2. Wiring of luminaires
3. Connection of portable luminaires, portable and mobile signs, or appliances
4. Elevator cables
5. Wiring of cranes and hoists
6. Connection of utilization equipment to facilitate frequent interchange
7. Prevention of the transmission of noise or vibration
8. Appliances where the fastening means and mechanical connections are specifically designed to permit ready removal for maintenance and repair, and the appliance is intended or identified for flexible cord connection
9. Connection of moving parts
10. Where specifically permitted elsewhere in this Code

At a stretch, if you could permanently attach the transformer, you could maybe consider this to be under 400.7(A)(8), but then you're still required to have a removable plug:

(B) Attachment Plugs. Where used as permitted in 400.7(A)(3), (A)(6), and (A)(8), each flexible cord shall be equipped with an attachment plug and shall be energized from a receptacle outlet or cord connector body.

Additionally, I think this could be interpreted as violating 400.8:

400.8 Uses Not Permitted. Unless specifically permitted in 400.7, flexible cords and cables shall not be used for the following:

1. As a substitute for the fixed wiring of a structure
2. Where run through holes in walls, structural ceilings, suspended ceilings, dropped ceilings, or floors
3. Where run through doorways, windows, or similar openings
4. Where attached to building surfaces Exception to (4): Flexible cord and cable shall be permitted to be attached to building surfaces in accordance with the provisions of 368.56(B)
5. Where concealed by walls, floors, or ceilings or located above suspended or dropped ceilings
6. Where installed in raceways, except as otherwise permitted in this Code
7. Where subject to physical damage

So I'd say, no, you cannot just hardwire the transformer in any way.

Additionally, you need to ensure the transformer (and all connections) are accessible for servicing. Transformers also get hot and need cooling - failure to have adequate cooling could cause the transformer to prematurely malfunction and/or lead to all the other problems of hot things in contact with flammable materials like your walls.

I'd find some area (such as under the stairs) that is accessible and unfinished, and run a new receptacle there. Then you can plug in the transformer, run the low voltage wires back to the LED strips, and you're good to go.

It's quite common to remotely mount transformers -- eg, it's very common for low-voltage under-cabinet lighting in kitchens that the transformer is located in the basement near the electrical panel or other unfinished (serviceable) area.