After much frustration in trying to find a replacement for the dongle receiver for my HP Wireless Elite Keyboard/mouse combo, I decided to pull it apart and see what was inside. My dongle was damaged when it was bent slightly because it sticks out of the side of my laptop and soon after stopped working.
I did the following to repair the receiver and get it working again:
1. use a small screwdriver to gently pry the windowed cover off the dongle. There are glue points at approximately each corner of the dongle.
2. Open the dongle to reveal the circuit board and you will see that the male USB connector joins to the little circuit board via 4 distinct connections that are held in place with solder. When the dongle gets bent, these connections are the first to crack. In my case, two of the four were damaged.
3. Using a small solder iron for electronics and some solder paste, I was able to very carefully apply enough heat to each connection point to repair it. I added a very small drop of additional solder to each connection, making sure not to touch any other components on the board.
4. I tested the repaired connections with success and then used hot glue to apply re-enforcement across the four connections and at the base of the USB connector. I then replaced the board in the cover and glued it back together.

Since HP will tell you to go buy another Elite keyboard and mouse and you cannot order a replacement dongle, this is the only way to avoid scrapping your entire wireless combo.