Apple created the MagSafe connector to prevent people from knocking their laptops on the floor if they accidentally tripped over the cord. It is a very sturdy little connector, but in this case, the wires had gotten loose inside the ferule end of the connector.
Since the wires were loose inside the connector, I decided to make things a little worse before I made them better. I pulled the wires out to get a better look at what I was working with. The wire inside is just a 2 wire coaxial cable.
Step 1: Disassembly
Step 2: Reassembly
After I striped the outer jacket off the power wire, I saw that this wire is made like coax cable. It has a mesh of wire outside another fully-insulated wire. I pulled the wire mesh out and twisted it up to make a regular wire. I decided to heat-shrink some insulation on the bare conductor to avoid any shorts on the aluminum ferule when I put everything back together.
I tinned the wires to make it easier to solder them to the end of the connector. With the connector in a helping hand clamp, I soldered the wires to the end of the connector. I then plugged the power supply into the wall and used the voltage meter again to test the voltage. According to pinouts.ru, the voltage should show 6.86 VDC under no load, which is what I got when I touched the leads to one inner and one outer contact on the connector.
I put a little piece of electrical tape over the exposed electronics on the back of the connector and put the ferule back in place around everything. I squeezed the ferule back around everything with a set of pliers, but there was still a gap at the top and the back where I had cut away the metal.
Step 3: Clean-up
I used this white putty that is a little softer than the black, plumber's putty that I usually use. I figured, since it's for a Mac it should be white. Unfortunately, there was no easy way to leave the LEDs exposed.
I gave the repaired adapter back to my friend, and everything is working perfectly!