How can I mount an 12V transformer for LED lighting inside a wall?
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:
  
  
  Pendants
  Wiring of luminaires
  Connection of portable luminaires, portable and mobile signs, or appliances
  Elevator cables
  Wiring of cranes and hoists
  Connection of utilization equipment to facilitate frequent interchange
  Prevention of the transmission of noise or vibration
  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
  Connection of moving parts
  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:
  
  
  As a substitute for the ﬁxed wiring of a structure
  Where run through holes in walls, structural ceilings,
  suspended ceilings, dropped ceilings, or ﬂoors
  Where run through doorways, windows, or similar
  openings
  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)
  Where concealed by walls, ﬂoors, or ceilings or located above suspended or dropped ceilings
  Where installed in raceways, except as otherwise permitted in this Code
  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.