You should be able to repair it as long as the faucet housing/copper tube did not get bent.
The directions for removing the stem (courtesy of Woodford)....
"Valve stem assembly (operating rod) removal.
To remove the operating rod from the faucet, shut off the water supply to the faucet and remove the handle and packing nut. Use the handle to turn the rod counterclockwise until the rod becomes disengaged from the valve seat. Use a small screwdriver to pry the packing loose. Pull the rod out of the faucet. The rod may have a check valve that will add resistance to removal, but a swift pull will disengage it."
Of course you will have to use pliers instead of the handle since it was broken off.
Measure the overall length of the stem (including the piece inside the handle) and cross reference it with the lengths in the chart to find out which length of faucet the stem is for. The faucets are labeled according the the wall thickness the faucet is made for, not the actual length of the faucet like some other companies list theirs.
Measure the faucet without disassembling it.
If there is anyway you can measure the length of the faucet from the back of the plastic flange to the tip of the threads, then you wouldn't have to remove the stem to know the length of the faucet. Subtract 1-1/4" from that measurement for the correct size of faucet that you have.
Now just go to the supply house and purchase the proper length valve stem assembly for your faucet. Be sure to check the price of the complete faucet against the price of the repair parts. It may be more economical to purchase the complete faucet.