Kane won the Ballon d'Or last week, three days after Kompany collapsed in training and was rushed to hospital. Kompany, who was diagnosed with meningeal irritation, was released a few days later and is expected to be sidelined for some time.
Bayern will also be without Ismael Saibari and Serge Gnabry, who were injured in their 2-2 draw against Borussia Dortmund in the league on Saturday, with the former undergoing surgery.
Jamal Musiala and Luis Diaz are also still out, recovering from their injuries.
"It was a bit of a rollercoaster of emotions on every end," Kane told a press conference ahead of Tuesday's match in Madrid.
"A big high for me personally last week, some lows with the boss and the injuries. A lot of extremes. We had to deal with that, and it brought us closer together."
Kane, who said he was "extremely close" to a contract extension, said Bayern would need to maintain their focus until the international break later in November, during which they expect some players to come back from injury.
Kane missed a penalty and hit the post in the draw against Dortmund, his first match back since his Ballon d'Or win.
"For now it is fight for every moment. We are missing some key players and our coach but it gives everyone an opportunity to prove themselves in this team," Kane said.
