LEDs stay on (very dim) when switch is off
We see this a lot.  "I don't have a dimmer" -> some oddly reluctant exploration eventually reveals they have some other device with the same effect.  

the light kit has its own remote-control switch -- you just answered that yourself.  That remote control switch isn't fully turning the light off.   Since you mention LED strip and remote in kit, I assume this is one of those very common LED strip controllers that lets you select 33 colors, rolling colors, all that.  The kind that costs $6 on eBay, which explains everything.  

Those controllers' real purpose is dimming/color control.  They could be built to provide a hard off, but clearly this one isn't, and in any case, the DC power supply is still on.  

It really needs a separate hard switch, and not be left on 24x7.  Many of these products are cheap in a way only one nation can do, and you don't want one of these running unattended, catching on fire and filling your house with toxic smoke. 

Since the other light is also an LED strip, you may be up against the same problem, just in a different way.  If you start measuring various points for AC and DC voltage, bet you'll find voltage past a point where you shouldn't.