If Spurs actually did win the EL, I think lifting a trophy would be enough to keep Kane on board for at least another season.
On a completely unrelated general note, this is a difficult topic for me. I started watching Spurs religiously because of how much I love Kane (my only real team, as such, is England), and I've developed a soft spot for the club over the last six years. But ultimately I want Kane to have the career his talent deserves more than I want Spurs to win something. If they happen to coincide, fecking fantastic, but there's only one winner in my mind if those two priorities conflict.
I hope Kane doesn't give Spurs too many more of his peak years if there aren't real signs of very significant improvement in the near future. He's just too good to end his career without a single trophy, unless he wants to go for the oddly distinguished honour of being the greatest player of all time never to win anything, which he undoubtedly would be if he did go trophyless until the very end.