Furni Gator SE Alarm Clock - Rebuild

My wife found one of these at a thrift shop. The LED display was missing segments so I took it apart:

The side of the clock has got a button panel that looks like this. I'm hoping to leave it as-is, and simply wire the buttons to the input pins on my Arduino:

I managed to get power to the Arduino through the existing 5V power supply. (Yes, I know I have the red and black jumpers backwards; I soldered the wrong colors onto the leads...)

So far so good, I can light up an LED:

Where I'm stuck is getting the buttons for the clock to register input on the Arduino. I used the simple example script where you hook your button up to pin 7 and use it to turn the LED on and off, but I'm not registering any input. Any ideas how I can make this work? I'd like to use that ribbon cable coming off the button panel rather than soldering each switch independently, if possible.

I'll post the labels on each button, and each of the wires from the ribbon cable later this evening.