A play in for the sake of a play like how the NBA does it is incredibly stupid. However, I don't hate stealing an idea from the CFL and making it fit the NHL, the cross over.
First off, ditch the awful divisional crap and go back to 1v8, have the 2 division winners seeded 1-2 if you really want to. Then have a rule where if the 9th place team in one conference has more points than the 8th place team in another conference, have a cross-over, or even better, a 1 game play-in. In this case, it would've been Calgary vs Montreal for the right to 8th place in the east.
Since I'm sure people will cry travel and how unfair this is to the poor eastern teams, then give each team a choice. You don't have to go full women's hockey where they pick the opponent, but let the #1 seed decide between the cross-over #8 or the in-conference #7. And then work down the list until #4 is possibly forced to play the cross-over if nobody else wants to.
So in this year's case, we do a 1 game playoff. Can even let the 8th seeded lower ranked team have home ice since they're "defending" their playoff position. In this case Calgary @ Montreal. Lets say Calgary wins. Washington gets a choice, #8 Calgary or #7 NJ. They pick NJ because of travel. #2 Toronto then picks #6 Ottawa over #8 Calgary for the same reason. #3 Tampa then has the choice of #8 Calgary or #5 Florida, with #4 Carolina getting whichever team is left.
Let's say Washington (1), Toronto (2), Calgary (crossover), and Carolina (4) win their 1st rd series. The highest seeded team, Washington, can again get their choice if they want to avoid travel, so they could pick between Calgary (western crossover) and Carolina (other team that wouldn't have home ice in rd 2).
Since the teams picking their opponent isn't supposed to be about strength of opponent, but rather travel, the 1-4 seeds should be required to submit their choices by say 5PM on the day of the play-in (before the game). Washington writes in their choice of Calgary or New Jersey, Toronto writes Calgary or Ottawa, and so on. If Washington picks Calgary, the other 3 teams' choices become void. And if Montreal wins the play-in, all the choices also become void. If there is no crossover, then the seedings stay as they are.