「うん、もうすぐ[寝]{ね}るから。」
「から」 here is used like a sentence-ending particle, and that is one very common usage of the word in informal speech.
We use 「から」 this way to make an announcement and see how the other person would react. More often than not, the speaker simply expects that reaction to be along the lines of 「わかった」、「じゃあいい」、「それならいい」、etc., which would be the Japanese equivalents of "Alright, then!" or "Okay, then!" In other words, this 「から」 is not really used when one knows that one's statement would cause a further argument.
"Right, I'll hit the sack in a minute!"
In this particular line by Maruko, the nuance of 「から」 would be like saying "Don't you worry!" without saying it.
In other situations, however, the から-ending can be used to make stronger statements of warning, determination, etc. as in:
「[死]{し}んでも[許]{ゆる}さないから!」 "I won't forgive you even if I die!"
「ママに[言]{い}いつけてやるから!」 "I'll tell Mom on you!"
EDIT: Similar example sentences can be found in Definition #3 in this dictionary as well.