Is a 15-minute private lesson worth it?

I don't think there are private instructors that teach just for 15 minutes. Many that I have known tend to teach for 20 to 30 minutes. If you go to a local ice rink, ask any of the teachers you take your group lessons from if they also teach private lessons and are willing to teach you. And talk to them about prices and time. You might be able to find a teacher who's willing to teach your 15 minutes at a reasonable price.