I don't think real Madrid really needs to sign a new player in other to play very well in next season, they already have a lot of good players in the team the only thing they lack is team work and a good formation that will make the players function very well and I think that is what there New coach Xabi Alonso is doing, if you check they way they are now playing in the FIFA club world cup you will see that there's a much difference, and I believe from next season they will do more better.
Real Madrid is a great club and for last season I won't say they failed they played very well last season and where in a good position in the la Liga table and also tried in the champions League competition, the only thing that they failed to do last season was to defeat Barcelona in all the matches that Barcelona and Real Madrid played Barcelona won them and that was because they didn't play like team and if Xabi Alonso can make them play like a real team then they will become more stronger.
You are right, teamwork and also playing formation are indeed one of Real Madrid's obstacles that made them play less than optimally last season. So if Real Madrid can overcome all of their shortcomings, of course they can definitely return to their good performance again. But even so, I think apart from those two things, what Real Madrid needs to fix right now is the midfield. Because in my opinion it is still undeniable that Tony Kroos' departure made Real Madrid's midfield a little less than optimal in flowing attacks. So right now Real Madrid may need to fix that too, either by training the existing midfielders to be better or bringing in midfielders who can make Real Madrid's midfield good again.
In addition, in the World Cup between clubs, Real Madrid's performance and playing style were indeed a little different. But even so, in my opinion there is nothing special about the change. Maybe this is because the players are still in the adaptation stage, so all the strategies given by Xabi have not been completely smooth. Maybe in the coming season, Real Madrid can indeed be better with the strategy implemented by Xabi. So basically I pray that everything goes smoothly for Real Madrid.