The K-240 Monitor work fine with a stationary amp, even off a mixer's (or monitor speakers') headphone socket. A CMoy amp doesn't have the same amount of current available, and hence it doesn't even make the headphones cut external noise off as well with the same kind of music. It just can't drive the K-240 to the same volume/frequency coverage. Harmonics sag, dynamics lack, it's not a good lively full-range frequency coverage. It's that simple, and the K-240 being a serious power hog, it cannot be driven by anything less than pro gear or a stationary stereo system amp's headphone output.
The newer AD8397 CMoy amp isn't specifically meant to drive K-240 Monitor, though that'd be a nice plus. It's the DAC headphone out that has to be capable of driving 600-ohm headphones, and it'll still need a step-up stage (8-12V off USB 5V). As it's been said, Wakibaki has an interesting AD8397 circuit, and there's a Swedish DIY fellow's AD8397 design too which ought to work. Both are rather complex, which is a minus as the cheaper and simpler the DAC output stage, the better, but hey, looks like that's the tradeoff of using AD8397.
Current measurements aside, just stacking a few AA rechargeables ought to do the trick of measuring how much current the K-240 Monitor really need.