Add a standard USB cable to your PS Vita dock to free your Vita USB Cable.
You can charge and use it on the PC too :D

What you need:
-PS Vita Dock
-Usb Cable (You can Scavenge it from anything like old mouse/keyboard...)
-Soldering Gun + Solder
-Small Phillips Screwdriver
-Drill or Xacto to make the hole
-Hot Glue (Optional)
-"C" Clip to limit the wire (Optional)

If you want to have another cheap wall usb charger you can buy PSPGo charger they are on sale
almost everywhere. It's the same spec. as the PS Vita one. (I pick 2 at Staples for 1$ each !!!)

Step 1: Disassemble the Dock

1- Unscrew the 3 black screws under the dock.
2- Remove the Center Part.
3- Remove the 4 silver screws that hold the board.

Step 2: Make the Hole

Make the hole between the 2 existing one.
The size depend of the gauge of your wire.

Step 3: The Wire in the Hole...

Pass the wire in the hole and install the clip.
If you want to glue it instead, wait after the soldering to get some freedom to work.

Step 4:

Your cable should have 4 to 5 wires colored with Red, White, Green, Black and a Shield
If not you need to find which is which, go online and check USB Pinout to find out.
Solder your wire like in the picture, 1 and 2 can be soldered together, 14, 15 and 16 too.

Step 5: Always Check Twice !!!

If you have a multimeter, check that you have everything in order and you
don't have soldered 2 pin together !!! (except 1+2 and 14+15+16).

I don't think you can broke the PS Vita with those voltage, but better be safe than sorry !

After that you can clip your PS Vita and plug it on your computer to test.