Did the piece on the adapter break? As in the small metal tab that sticks out of the adapter? Or did the endpiece in the picture break off?
Assuming you are talking about the piece in the picture, that is a female quick disconnect terminal. They come in varying sizes, most likely you will want one for wire size 22-18 AWG (wire gauge).
You can buy them at any Home Depot, Lowe's, local mom-and-pop hardware store, Radio Shack, etc.
To use, you insert the wire (after you have stripped the shielding from it) into the small opening at the base of the terminal. You then crimp the lower half of the terminal to secure the wire in place. You're done. Gently press the new adapter onto the male ends protruding from the solenoid.