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:
- 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
- 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.