Finally found some time to sit down with all the bits and assemble a second Morse code key using the supplied schematic. At first it didn't seem to work, but after checking through everything I found one of the speaker wires wasn't fully inserted (short lead). After I fixed that, it buzzed at me when I clicked the button! The button is the clicky kind, so it's not that well suited for doing Morse code, but it works! The buzzer was also a lot louder than the kit, which I later discovered was due to me forgetting a resistor between the final capacitor and speaker.
The next step was to make it light up in addition to buzzing. Spent some time looking at the schematic for a place to insert an LED that I didn't think would make it go poof or fry anything. Put in a red LED and connected it to ground, pushed the button and it lit up! Then I decided to try a second green LED in series with the red, and they both lit up! Success!
A variable resistor instead of the fixed resistor in front of the speaker I think should give me some volume control, so I'll try that next. I need to get one first.