Here is the back of the BaB board, with the two cables running to the speaker it's originally attached to. Take these off with your soldering iron, and attach the red wire coming from the EB speaker to these points (it doesn't matter which you connect to which point).
Here is the front of the BaB board, and the two highlighted points are where you need to connect the two wires you previously attached to the EB board to. Again, either cable can be attached to either point. These points are a bit closer together though, so do be careful when soldering them.
Fit the board in the area you cut out. It does take a bit of fiddling to get positioned properly. Make sure you leave the two buttons and the microphone in the open area so when the battery cover is removed, you can get to all three easily.
Once you've done this, screw down the plastic portion you removed earlier. After that, screw down the EB board, like so:
Put the outside and the top back on it, and make sure you line up these notches with the cutouts:
for a solid fit.
Finally, check out the job you did by removing the battery cover.
The first highlighted thing is the light that goes one when you are recording. Second, is the play button. Third, the microphone, and last but not least, fourth, the record button, hidden a bit in my placement, though still accessible.
Record something, then press down on the top and hear your newly modified Easy Button.