So I'm no expert and I'm not advocating dumping the whole roster, but I think the main way they can fix their toughness/drive/effort issue is by getting younger.
I'm of the belief that a coach's system is the #1 factor in how well a team performs at certain aspects of the game more so then roster makeup. Sutter runs an overly conservative suffocating defensive system in LA that allows them to fill in young players and not miss too much of a beat as the system covers up for most developmental warts for the most part. Jeff Carter helped 2 rookies perform on par with most of our wingers last playoffs, and he's not the calibur of Crosby/Malkin in that regard, but the system is so structured and sound to allow them to always be in the right positions for proper defensive coverage so they can take advantage of more offensive turnovers or mistakes. And even tho they play that overly defensive system, they can still run and gun when needed as evidenced by all the high scoring games they played last post-season.
Since Johnston's system is not as dedicated to sound defensive positioning as LA's or even CHI's, they'll need to rely on speed, outworking the other team to maintain puck possession, and consistent effort for 60 mins, all of which they've struggled with at different times this season and in the 5 prior. To get back to my general point, the Pens visibly seemed to be faster, stronger on the forecheck/backcheck, and more consistent in general effort given before they won the cup. And while not every important piece was young back then (Gonchar, Scuderi, Gil, Sykora, Guerin, etc.), they just visually seemed to play "hungrier" more consistently when the core of the team was still mid to early 20s kids who hadn't won anything yet. You'd think that's an obvious statement but maybe it's more important than we think.
Obviously recent cup winners have a mix of old and young so that's not the be all end all solution, and you obviously have to have great players regardless of age, but could maybe part of the perceived "soff" issue be that they have relatively too many "older" players who have families and a paycheck to worry about than younger ones who can go 110% all the time with less in their lives to worry about outside of hockey? Bigger chance I'm off the mark than on it, but Shero's plan to sacrifice picks for deadline rentals most years I think hurt them in more ways than just prospects, they just don't have that youthful drive anymore, like the cliche of it being a job for a majority of them now instead of a game they enjoy.
Just my thoughts, again I'm no expert Mr. Sutton.