How do I fix this wiring to properly run LED lights? (More details/products added)
Romex is not supposed to just stick out of the wall.  Any in-wall wiring is only supposed to connect to out-of-wall wiring through a junction box.  The cleanest way to do that is to put in an electrical outlet.
You can get retrofit junction boxes that just hold on to the drywall, which makes it quite easy to add an outlet.

(source: homedepot.com)
Also, those power supplies may be way overkill.  How much lighting are you planning to install?  I just installed about 8 feet of strip LED lights and it's using about 2 watts.  If the power supply is over-rated by 10 times, it's wasting power just sitting there turned on.