You have quotes inside your quotes. Try the following:

print $sh->ls( qq(-l ./logs/ |grep -v "^d" |grep -v file1 | awk 'NR>1 
+{ print \$9}'));
[download]

Or indeed you could use q() as you're not doing any variable substitution inside the string.

Update: yes I know it would be better for the OP to have coded this in perl rather than shelling out 4 subprocesses. Sometimes you need to start with where the person is. Spotting the quote nesting is something glaringly obvious to me - a kind of mistake anyone can make in coding, and worth pointing out.