Here is a list of suggested tools and materials you might need:
Soldering Iron, Solder
USB Compact Flash card reader, $2 online (slightly smaller than the AE so it fits inside)
3 wires (2-3inch in length), one for ground, one for 5v and one for 3.3v.
Update: I changed the blue tape to white tape on the reassembly of the housing.
Step 1: How to Open the Airport Express Case
Step 2: Disconnect the Power Cable
Step 3: Test the CF Memory Card Reader
You can use your multimeter and test your power supply too. A word of warning, there is a risk of electric shock if you do decide to tinker with your broken AE power supply.