So I finally got a peltier cooler from ebay last week. You can buy one for about $5 (depending on postage it could be more). I dropped some water on it while being connected to my PSU (14.5V, 23A) and it froze almost instantly. However, when I tried a larger amount of water (13 mL, a volume of one ice-cube), it never froze. The problem is in the cooler block on the other side, because it's not efficient enough. The whole system overheats after a couple of minutes.
Here is a picture of the setup of the first experiment (PSU not visible):
(click to view in full size)
And here is a temperature/time plot of the experiment. Note that temperature is in degrees Celsius.