First, remove the silicone tube from around the LED strip. Insert diagonal cutters in the end and cut away the end cap and at least an inch of tubing. Then, carefully remove the whole end cap and separate the LED strip from the tubing. Repeat at the other end. With the 1 meter length, it is now possible to carefully pull the strip out of the tubing. With the 5 meter length, it is probably necessary to slit the whole length of the tubing.
Separate the strip into 250mm sections by counting off 10 of the divisions. Make a clean cut exactly along the dividing lines using diagonal cutters.
Next, for each section, cut a generous length of red and black wire (4 inches is good, but I probably did about 6 inches. This leads to some waste but it's preferable to ending up without enough wire). Slightly strip one end of each wire (even though I keep my fingernails short, they were a good tool for this task; the traditional wire strippers I own were not "up to" dealing with this small and thin wire).
Desolder any original wire (it is way too chonky to be good for this project) and then solder your new wire on: Red on the "12V+" side, and black on the "12V-" side.
It's best to leave the other end of each wire unstripped until soldering it to the Perma Proto board.
Now it's time for the part that needs a lot of care: inserting the LED strip into the 3D-printed LED guides. Identify which side to insert the strip from, and which direction the LED strip needs to face. You'll need 6 strips inserted from each direction; the strips should all face "out".
The first part of the strip will insert easily, but after that there is increased friction. Hold the strip between your index finger and thumb, with your thumb on the plain side of the strip. Push it forward by short distances (a few mm / ¼ inch at a time), being careful not to kink the thin PCB material, until it's all the way through the guide.
While this picture shows some of the original wires, I discovered that these were not flexible enough, and replaced them with fresh wires.