Clearly he wasn't too close with Lewandowski or Gotze.
Nobody was ever too close to Lewandowski. The Pole is the prime example of someone who sees his career as a profession and nothing else, which is completely ok. He never really got attached to the club, the team or the coaches. He was a lone wolf who delivered on the pitch and is still respected for that by the supporters but was never particulary well liked inside the squad or by the supporters.
Now Götze is a whole other matter. Klopp was clearly emotionally attached to him. He integrated him in the first teams training at 16, gave him the first minutes at 17 and made him a starter with pretty much a free role in the offensive midfield a year later. He furthermore went to great lengths to develop him in a calm enviroment away from the hype and expectations the media unleashed on him, including building him up very carefully after his serious injury in the first half of 2012.
Must have felt great when the boy declared internally that the main reason for his transfer to Munich was the coach. Out of all angles Götze could have chosen to justify this transfer, a lot of them being valid reasons (joining the best team on the planet back then with a higher prestige, better chances to promote his personal career or the pure financial aspect) he went for the one which hurt Klopp the most. The once in a lifetime opportunity to play under the coach he always looked up to and saw as the ideal coach for him. I don´t care how much he appearantly admired Guardiola, Klopp did not deserve to be basically told to the face that he was not good enough anymore. When Klopp explained that with a pained look on his face in the press conference before the Madrid game in the CL the feelings among supporters towards Götze really shifted from shock and sadness to anger. Different to popular belief, the main reason for the enourmous antipathy towards Götze was not the transfer itself, but the way the whole thing was handled by him.
Coming back to Kagawa, it can´t be denied that the Japanese formed a very strong partnership with Lewandowski at the end of his first stint with us. That was never the case with Götze, though. Their repective individual abilities allowed them to play together well enough, but there was never any special synergy they shared. Infact they had their strongest spells when the other was either absent or out of form.
In theory Kagawa finds the ideal circumstances to revive his career in Dortmund, which is why he took a pay cut and turned down offers from Spain and England. He is back in his comfort zone playing under a coach who clearly rates him and for fans who love him. That alone will be a huge boost for his confidence. Kagawa could very well be a player, who needs to really feel the reassurance and trust from the club/coach to perform well and his inability seize his own destiny without it is probably the one big thing you could have fault him for at Manchester.
In terms of teammates the vast majority of the midfield is still there. Lewandowski is gone, but for him they have bought the closest thing in the Bundesliga in terms of play style (albeit with less overall quality) with Ramos. In Reus, Immobile and Aubameyang he finds several excellent targets for his through balls and key passes.
The partnership I´m the most excited about is with Henrikh Mkhitaryan, who is becoming more and more of the key player the officials hoped to be. They share several traits: They are very direct players, both in terms of passing and movement and the Armenian is one of the few offensive midfielders in the world who can at least match Kagawa´s pressing strength (an ability which was completely wasted at United). They could form a pressing unit which could be very hard to overcome. Furthermore, I think they could compliment each other very well. Kagawa is more technical on the ball and can work very well in tight places, which binds opposition and creates space for his teammates. Mkhitaryan has his strengths more in his tempo dribblings and shooting, which work the best with a bit of said space.
That is all only on paper, though. It will come down to Kagawa himself to capitalize on this second chance he was given. Klopp will have more patience with him than many others, but he still has to deliver and prove his worth again. It is not that common that a player gets such a good shot at redeeming himself on top level. A lot of it comes down to the emotionality of this transfer and United´s unability to negotiate hard.