I was able to get my left analog stick press working again by using a Contact Cleaner (or TV Tuner Cleaner) spray, such as you can find at Radio Shack.
Open the controller and remove the circuit board. Disconnect the motors and pull the plastic thumb stick heads off. Pay close attention to the micro push button on one side of the analog stick housing. This little black button is clicked by that white stick sitting above it. Even if it clicks the button down, the contacts inside of the button could be dirty, which is where the Contact Cleaner comes in.
Then take the board and the Contact Cleaner outside (the fumes are not good to breath in). Spray a small amount on that switch and on the analog stick for good measure. Be sure to move the stick around and click the button a handful of times so help the cleaner work its way into the contacts. After allowing it to dry outside for a few minutes, take a towel and wipe off the excess residue off of the rest of the circuit board.
With any luck the button will be working for a good while. This won't work every time as all buttons and switches wear out over time and need to be replaced. Serious repair technicians will tell you this a temporary fix. If you are inclined to do some soldering, now would be a good time to order a new Analog Stick control*.
This process took me about 15 minutes.