The first thing that needs to be done is solder the headers to the components. I would suggest putting the headers and the module in a breadboard while soldering to keep them perfectly aligned. However, be sure not to apply too much heat for too long or you could melt your breadboard.

A good solder joint will create a cone around the pin and a circle around to hole, has no gold from the board showing, and prevents the pin from moving. Press the iron and the fluxed solder near the bottom of the pin. If you're getting clear/yellowish flux everywhere, then your soldering iron isn't hot enough. In the case of non-adjustable ones, you probably just need to wait a little longer for it to heat up. Keep your area well ventilated.

You'll need 12 on either side of the Arduino Pro Mini. Solder from underneath with the long end down (long end and black strip are not on the size with the button). Now you need to solder the I2C lines. They are inconveniently placed in the middle of the board. Solder about two inches of wire to all four pins. We'll only need A4 and A5, but the other will help if you decide to do something else later (in my pictures I didn't realize this and had to balance the wires in the holes while testing on the breadboard).

Next solder all the headers to your Logic Level Converters.

Don't solder headers to the compass module! I wanted my screen to sit in the top center of the board, and the only way to do this without putting anything on the other side was to stick it on somewhere else where the pins don't lines up. Instead, solder a very small amount of jumper wire to the top of it. About an inch will do. Cover the bottom of the breakout board with electrical tape to make a small space in between the two boards when they site on top of one another. You could also use double sided tape. My picture has headers on it while testing because I didn't see the problem yet. I desoldered them later.

Now, on the screen board, solder headers, with the long part and black strip facing down, to the row with the square GND on it. I would also recommend soldering one header at one edge of the other row for stability and to help secure it to the protoboard (I didn't realize this until mounting it permanently, so it's not in the picture). On the other side we'll need clearance for the screw hole. I would not recommend soldering right under the screen's fragile connector for risk of damaging it.

You're now ready to put it on a breadboard and test it!