The LDR is not only a highly non-linear light-dependent resistor, it is also slow to recover to its "dark" resistance after exposure to a bright light that drops its resistance down into the k-ohm range. The "recovery" time for full dark resistance can be several minutes for some devices. This delay in dark recovery can mislead you into believing the threshold adjustment is incorrect, so you change it.
Because of the wide swings in LDR resistance, I would buffer it with an emitter follower. If you use a 10K potentiometer inserted as the emitter resistor, you can adjust the threshold input to the Schmitt trigger with the potentiometer wiper. I would also increase the resistor in series with the LDR to 100 kΩ so that about half the supply voltage appears across the LDR when it is dark, decreasing to almost zero when the LDR is illuminated.
I have a couple of LDRs that came with the Radio Shack Electronics Learning Lab Kit I purchased last year for one of my grandchildren. Since he lives in Virginia Beach, VA, and I live in Dayton, OH, when I got back home I purchased the same kit for me to "play" with in case he needed any help. It is still in the original box, unused, since I haven't heard from my grandson yet.
So, if I can find the time, I will dig out that LDR and breadboard the circuit. Meanwhile,
@KTW will probably have solved his problem... .<sigh> ... and I can get back to messin' 'round with PICs and Arduinos. Interesting thread, though. The two-transistor Schmitt trigger is a classic (Google it) so I don't understand why it doesn't work with the LDR yet.
BTW, inserting both leads of the LED into the same in-line common tie point is an easy mistake. I've done that more times than I care to remember, but my excuse is old eyes.