How to connect an AND gate with a shift register
There may have been a wire in the wrong place or something. I did have it spread across three breadboards. I sketched everything out in a schematic, then pulled all the wires out and started over, following the schematic. Then things worked! Here's the schematic I used. The LED only lights up once the D input to the first flip-flop has gone HIGH and then LOW, which is how it should work.