How to create a song a hit

@TobasiHD Hello!  I'm not sure but it sounds like you may have been writing lyrics instead of writing songs.  You can only license lyrics once per sim week and "write lyrics" does not count toward the aspiration. To complete the "write 4 songs" part of the aspiration, make sure you select write song on any instrument once your sim has level 8 skill on that instrument.  From what you said, you should have this option for Guitar and Piano.

To answer your question about licensing, once you finish writing a song on each instrument, you can license one song per instrument each week.  As an example, my sim has written a song on the Piano, Violin and Guitar.  When I click on the mailbox at home, I see the options for each of those instruments even though the options are grayed out because I've already licensed songs this week.  Sorry the image is so dark.  It's nighttime now in Sim land. 🙂

Hope this helps!