The first thing i did was make sure all the leds were in the holes properly, with all the little legs in one direction and all the long ones the other way. then i flipped the piece over and checked to make sure none of them weren't sticking out to far (or not enough). While im sure no one else will ever notice (or care), it would bother me forever.
Next i cut some 20g uninsulated soft copper wire into sections long enough to connect all the legs in one row. I just laid the wire next to the positive leads and added a drop of solder to make them stick. Copper wire loves being a support in my electronics projects as its easy to work with and takes solder easily. I soldered all the positive legs together in one row, then all the negative ones, making sure to leave a little extra wire hanging off the side. I checked each row as i finished to make sure (one last time) that there were no bad LEDs. I actually found one that got heat damaged from the soldering and needed to be replaced. Its a lot easier to do this now then later, so check often. That extra bit of copper off the end is used to connect all the rows so the 25 LEDs are essentially all connected to the power supply directly.
Some people would call this parallel, i would call those people nerds. Normally im all about wiring LED's in series, but because these are >blinking< diodes they need to be treated differently. If you add anything after the blinking led in series, that extra component will only get power when the blinking LED is on. I have used this to my advantage in other projects, using the blinking led as a cheap/easy timer, but here i wanted each light to blink independent of all the others. So parallel is a must.
At some point i clipped the barrel plug off the wall wart and stripped the insulation off the wires. These bare ends i now solder to the leds. Because the diodes are all in parallel i could solder these power lines anywhere and the unit would function. After plugging it in to the wall everything worked great.
Heres a video:
The last thing to do is clip all those unsightly legs from the back and put it someplace where a red light is needed.