Source: page 32, Ch.82 of よつばと！
My translation of the highlighted text bubble is,
Now, sleep quietly okay.
But I am confused as to why 寝てろ is used here as opposed to 寝ろ? Is it because sleep is a resultant state (something like 知っています or けっこんしています)?
After some probing I came across this but it doesn't answer the difference between usage of 〜てろ and 〜ろ form.
What I did come to know is, 寝てろ is a compact form of 寝ていろ which is imperative/command form of 寝ている. All in all, 寝ていろ is something like a command to sleep (or be in state of sleep or keep sleeping?) Why is the progressive form used here? What kind of subtlety does it add?