Thanks for all the information.
What output, if any, do you see in the Serial monitor?
Do you have access to a multimeter for testing?
Did your module come with a sample RFID tag, for testing?
Does anything change when you attempt to scan a tag? I believe it is common for these modules to only output serial data when a tag is detected.
If the Uno is only being used to transfer the RFID module's output to the computer, there is an easier way than using SoftwareSerial.
If you flash a blank sketch to your Uno, you should be able to connect the module's tx pin to Uno's RX pin (Pin 0). The serial data will pass through to the computer just fine.
The module's information page suggests that the current draw is about 60mA (if powered by 6V). Even at 9V, I suspect the module's current draw might be a bit high for that 9V battery.
The information page suggests that a 5V power supply is acceptable. In theory, you should be able to completely disconnect the 9V battery, and power the module directly from the Arduino's 5V pin, for testing.
It would be nice to confirm the current draw first using a multimeter, incase there is some hardware fault, but if you are brave you can probably just connect it up..
The whole "wires taped / wrapped together" thing is not ideal, but hopefully it isn't the source of your issues.