Main issues:
- no solid veteran leadership
- a complete mess with the offensive system
- a lot of injuries every year
- Gaudreau not performing to his level and role
- no consistency with the goalies
OK, working from the perspective that a) I'm trusting your analysis of the issues and b) no "trade everyone I like to Philadelphia for everyone I don't then take Comcast's money and run" jokes...
Step 1 has to be to investigate why there are so many severe injuries. It happens to every team, of course, but you don't put together a winning team if 2/3 of your game-day lineup is supposed to be wearing Monsters jerseys. If there are problems with the strength and conditioning teams, replace them. If the facilities aren't up to par, invest in upgrades. If the issues are with the ice plant, that is job #1. Player health and safety has to be the first thing you look into, because it has knock-on effects for everything, not least of which being not appealing as a FA destination. You might never overcome the draw of Toronto or NY, but if you can promise players that they will receive better care and treatment in Columbus than anywhere else, that will sway some people.
Step 2, make better decisions with coaching. Columbus' list of truly successful coaching hires is John Tortorella, end of list. That should simply never be the case. My first instinct here is that you want someone who has had success with the whole "we're a ragtag bunch, but damn it, we'll outwork you" mentality, and that screams Craig Berube. There are other possibilities, I'm sure, but Chief has a ring, which brings credibility, and he's a strict guy without being Tortorella/Keenan levels of asshole. Then, let him make whatever decisions need to be made from a coaching perspective thereafter. This should alleviate the leadership issue, had best help with the offensive system, and all other things being equal, can do good work with the goalie consistency issue.
Step 3: veteran leadership. Berube can help with this, because he has connections throughout the league, players who loved him and would want to play for him again. Do I think there's necessarily a deal to be had here, I don't know, but I would at least investigate the possibility of a deal with St. Louis featuring Johnny Gaudreau and Brayden Schenn, if you get Berube in town. Also, since I know Philly, my mind immediately goes to trading for Scott Laughton, who is generally looked upon as a good veteran leader, good in the room, even though he had a bit of an off year on the ice last year.
The goalie market is absolutely dire this offseason, unless you're going the trade route, so I would make resolving that a future year's problem, unless you can solve it by getting the Islanders to let Mitch Korn go.