Well, the resulting sentence, "I know Kate who(m) we play tennis together" is ungrammatical. You can't use "who(m)" like this; the "who(m)" clause has to be descriptive of Kate, which "we play tennis together" is not. For example, these are valid:
"I met Kate, who is Sarah's sister." "I met Kate, whom I liked very much."
Grammatically you can say "I know Kate, with whom I pla