Using the colour balance sliders is quick, but a little crude. A more advanced and accurate way to correct a colour cast is to use the Levels function. Again, it's best to use an adjustment layer for this if your editing program supports this feature.
At the top of the Levels window is a drop-down menu for channel selection. Open this, and select the red channel.
In the red channel, the right side of the histogram represents red and the left cyan. You'll see that most of the image information is crowded over to the right, due to the colour cast. We can go some way towards correcting this by moving the left slider to the right, up to the point where the image information starts, and also moving the middle slider to the right, to a point where there is roughly an equal area of black either side of it. Again you'll have to judge by eye how much is enough, but in this case moving it from the mid-point of 1.00 to 0.50 is a good place to start.
In the green channel the left of the histogram represents magenta, while the right is green. The histogram is fairly well spread out, but we need to move the slider a little way toward the left, to help balance the histogram more evenly.
For the blue channel, most of the information is over to the left, at the yellow end of the histogram, so we need to move the right slider towards the left up to where the information starts, and then move the middle slider some way to the left to balance the histogram. Try to position it so that there is a roughly equal amount of black either side of it. As you do this, watch the colours change on the image, and you should see the white balance coming to its correct value.
Using this three-channel method you should be able to correct any colour tint caused by unusual lighting, or add colour tints if that's what you want to do. As for my starter image, it now looks like this, which is a big improvement.