HP15C On Button won't respond

[damaltor]

Those pins with the shortened pins with no solder on the trace are purposefully disconnected.
you could use a wooden toothpick and gently (!) push on the pins of the chips which are soldered to the pcb below. if any pin moves (in respect to the pcb) the soldered connection has failed.
if you can use a soldering iron, you might as well just resolder all of the pins anyways. Use leaded solder if possible.