Like Craig, I am a little confused about your post, not sure exactly what is missing.
If I read your post correctly, this is the situation:
Your tow vehicle has a 7 pin round connector for the trailer. That connector is still okay.
Into that 7 pin round connector you have an adapter that converts the 7 pin round to the 5 pin flat. (the adapter has the equivalent of the 5 pin version of the left hand picture that Craig posted). You still have that adapter and it is okay.
What is missing is the lenght of wire that goes from the trailer to the tow vehicle.
Is that correct?
It sounds like the chunck of truck tire caught the wire as it hung down (good reason to not let the wires hang down) and ripped it off. It simply unplugged from the tow vehicle, but it snapped the wires off the trailer.
What I am not sure of is if there are any wires danging around the tongue of your trailer at all. You mentioned a "taped" spot. If this is a lenght of multi-colored wires that ends in a wapping of black tape then that is where you need to splice a new connector into. Unwrap the tape and you will probably find a bunch of butt splices that have been ripped out. Cut the old butt splices off, get yourself a new trailer side wire harness (available at most auto parts stores) and just reconnect. If you are lucky the wires will match in color, just hook color for color and you should be fine.
If there are no wires dangling around the tongue of the trailer at all then it is likely that the wires snapped off inside the metal tubing of the trailer frame. This will make life a bit more interesting.
There are too many possibilities on exactly how the wires could have been run for me to give you reasonable directions on how to fix. Some pictures would really help out at this point.