I don't agree with you that Real Madrid don't need Mbappe now. Because they have newly lost Benzema to Al Ittihad so they need to bring a new striker before the new season starts. If PSG give a signal like they will sell Mbappe this summer then Real Madrid would surely be in line to sign him right away. For now they are just waiting patiently to see how things will go.
As I said if they can't sign Mbappe now there is still no problem. There are many striker alternatives in the market now to fill the space in the lineup for next season. I don't know how much money they are eager to spend though because I think they might plan to allocate as much money as possible for Mbappe's demands and a possible transfer fee as a plus.
If Real Madrid fail to sign Mbappe, then surely a contract with Kane will be signed. Moreover, there was information recently that Kane was telling his Tottenham teammates that he would move to Real Madrid. Real Madrid, after Benzema's departure, needs a super scoring striker with experience. In addition, recently, not least after working with Mourinho, Kane has improved his assistant skills quite well, which could be useful when playing in conjunction with Mbappe.