Check your preferences settings. Go to LW's Prefs>iTunes>Importing & enable iTunes importing & press apply. Go to Prefs>iTunes>Sharing & make sure your name & sharing option is ticked. Once songs are imported into iTunes then they are definitely within iTUnes music library (don't get confused with the windows version iTunes.)
Keep in mind iTunes cannot import some types of mp3s. Notable are Ogg Vorbis mp3 format, & rm (not sure about anymore.) So just double-check that all the songs you've dowlded are within the iTunes library. For this reason I keep a backup mp3 player in Audion which is now free!:
http://www.panic.com/audion/download.html which does recognise Ogg Vorbis mp3's.
The LW playlist within iTunes is only a reference list to songs in your LW downld folder. But these should have also been imported into iTunes if importing is enabled within LW. So as suggested above ... just double-check that these songs are listed in the main iTunes library. I set my iTunes to organise via artist & album; see iTunes Prefs>Advanced & tick both ooptions for arrangement & copy files to Library when adding songs.