Roseola seems to be the answer to K's 'illness'. The description provided here is exactly like what she had: she was miserable for about a week before this rash appeared with a low-grade temperature, lethargy, and sleepiness. The rash lasted about 12 hours, then disappeared and she was back to normal shortly after that.

Poor baby.

We were worried it might be chicken pox, but her symptoms didn't fit, and the rash never developed that way.

She is back to her normal, happy, outgoing self now.