Most melodicas have the air-flow from the low notes to the highest. When you are trying to play those left-hand chords they are robbing all the air. That’s why you aren’t hearing the high notes. The piano isn’t really a good model for playing the melodica. If you have any experience with an organ it would be closer.
Polyphonic playing can be achieved, watch some videos of performances of Masa Matsuda http://www.youtube.com/results?search_query=masa+mastuda
One thing to think about is how you arrange the notes. Left hand triads can get pretty muddy. Try playing two and three note chords in the right hand with the melody note always on top. Then use the left hand to play single note bass notes.