Translation:František is looking forward to it like a little child.
Disclaimer: I am learning, too (and this is one person's opinion)!
First, I'd say that "jako" has appeared in the course so far primarily with the meaning of "like" rather than "as," though that may not be a problem.
But to my (US English) ear, "like" just sounds better. "Like a small child" or "like a little kid" sound right, while "as a small kid" sounds... odd. Sort of like part of the sentence is missing -- as if it should be "as a small kid WOULD."
You could try reporting your version if you get this exercise again, though. Maybe "as" was just overlooked.
The construction "František is looking AS forward to that AS a small kid" would not be an improvement. "František is looking forward to that AS MUCH AS a small kid (would)" does work as an English sentence, but I don't think that's a translation of the exercise sentence.
My reading of the exercise sentence does not suggest that František is a small kid. The English sentence, "As a small child, František is looking forward to that" would make that point, while "František is looking forward to that like a small kid" suggests to me that František is a man, and he is as excited about something as a little kid would be... a party, a super new movie being released, Halloween, etc.
But this is just my interpretation. Perhaps one of the Czech natives on the team will tell us if the Czech sentence can be interpreted as the English sentence, "As a small child, František is looking forward to that."