Somehow you got the inputs and outputs mixed up I believe and your also making little sense. Explain:
Pin 7 is an input and pin #10 is an output.
You had a ground connected to the common of the BCD switch, right? Remove that ground.
Connect the Common to (pin #10), an output of the ULN2003. Pin #10 is labeled O7 for output #7. Pin #7 is the corresponding INPUT.
Putting a high (+5) on pin #7 will put pin #10 to "ground", thus it's now the same as when the common of the BCD switch was grounded. This will read the BCD value or complement depending on the code.
Putting a LOW (ground or OPEN) just opens the common connection and all bits should be a 1. It should read 15 or 0 depending on the code.