In 2012 LA entered the playoffs as basically a write off. They then went on to dominate in the playoffs, playing three quick series in the west and one long series in the SCF. They mostly relied on Quick who had broken the record for highest sv% in a playoff run with a .946 sv% and a 1.41 gaa (not sure if that one is a record or not). The Kings scored just under three goals a game those playoffs (2.85). They went up 3-0 in each series and didn't look back at all those playoffs.
In 2014 they were the polar opposite. The had entered with 100 points and many people had them winning (it was considered to be between them, Chicago, or Boston). The then took the longest possible route to make it out of the west playing all 21 potential games. Their only short series was the Final. Quick had a .911 sv% and a 2.58 gaa. LA scored 3.38 gpg these playoffs and had multiple losing streaks in the playoffs (three in the first round and second round and two game streak in the third round).
Has any team won two cups in under five years before while being basically opposite teams completely?
Detroit is the first team that comes to mind. 1997 and 1998 they made it look easier, I thought. They were down 2-1 to Phoenix in 1998 if you can believe it. But I believe that was it for those 2 playoffs. In 2002 they had a heck of a time getting out of their series'. Down 2-0 to Vancouver. Cloutier lets a shocker in from center ice and then they carry that on to win in 6. Beat St. Louis easily in 5. They were down 3-2 to Colorado before winning and down 1-0 to Carolina in the final. Even in Game 3 they weren't out of the woods, needing triple overtime to go ahead in the series. They didn't make things easy on themselves, but I think the 2002 Wings were "supposed" to win it in the public eye more than the 1997 Wings.
I guess the best comparison would be the 1971 and 1973 Habs in this context. 1971 I would say the Bruins were assumed champs. They were down 1-0 and down 5-1 at one point in Game 2 against the Bruins. Won in 7. Struggled a bit against Minnesota, but were never down. Were down 2-0 against Chicago and 3-2. Won in 7. While you never counted the Habs out, this was the Bruins' era and who would have thought an unknown like Dryden pulls a rabbit out of his hat. 1973 with Montreal being the best regular team by far they had an easier time. Only being down 1-0 to Philly. So it is sort of a reversal like it is with L.A. who did better when less people expected them to win. Also for what it's worth, Montreal was never down in their series' in 1968 and 1969.