Ok.
You see the long thin piece of plastic that attached on the button of the earse. You should have 2. The second one is probably stuck in the silo of the pen (or fell out).
In my case, both of them broke. Here is a tear down picture of the pen tear down which I took, as mine was actually more than the eraser that was broken broken, but fixed it as an emergency pen.
If the peace is inside the pen silo, you need to separate the pen middle section (where you see a groove). It is sadly has a bit of glue used inside, so that the top part doesn't turn from the bottom and block you from magnetically attaching it to the Surface Pro device power connector. The way I have done it, is to twist it with a bit of force to break the glue. When I twisted it, I made sure not to apply pressure on the pen clip, as this will break, bend. So I hold on the top plastic part that holds the pen metal clip.
Once you got it free from the glue, you can now pull hard and it should come out. Be careful not to break the pen clip.
Once you separate them, you now have 2 pen parts as shown on the picture above. The circuitry is well inside each components. You want to get the eraser circuitry out. So looking at the inside of the eraser part of the pen, where you split it open, using a tool, you can push the inside. It will come out from where the top, where the button is. You should now find the missing plastic bit.
Careful: Notice the pick wire that forms a cylinder that comes out of the silo. the wiring is very thin and can break. Never pull on it or twist it.
You can now put back in place the pink cylinder in place (it doesn't hold great normally, so don't expect that), as in the picture.
Now, using crazy glue, A HINT of it, attach back the plastic clip that broke off, and that you retrieved. Be VERY careful to position it exactly where it is supposed to be attached on the pen button. And make SURE that not a trace of glue goes in the components INSIDE the pen button that dropped out, if it does, then it will be stuck inside, and the eraser will no longer work.
If the content of the pen button dropped out, you need to put it back the way it was exactly. Once you do, rebuild the pen and make sure it works correctly, and disassemble the pen again, without loosing the button order. From memory, they are 3 components. A spring, a white plastic cap, and what it looks like a mini ferrite core that looks to be the same material from what the pink wire wraps around of.
Once glued back, pay careful attention that the glue doesn't go inside the pen button. If you used crazy glue, it should dry within 5min or so, but I would wait extra time to make sure it is fully dry. See glue instructions.
Once dry, if you have too much glue protruding outside, you want to use sand paper or something to scrap it out. The pen doesn't have much wiggle room, so it will be blocked inside. and the eraser button won't come out on its own once pressed.
Once done, you are assemble back the pen. When you do, and ready to slide in the button so that it clips again and forming your pen back the way it was, look inside the silo of the pen first, you'll see 2x groves, for the plastic peace of the pen to slide in. Position the button accordingly, and slide it in. Your pen should be be fixed now.