解:$(2)$对于$y = kx + 2k,$令$x = 0,$得$y = 2k;$令$y = 0,$得$kx + 2k = 0$
又$k\neq 0,$所以$x + 2 = 0,$解得$x = -2$
由题意,得$\frac 12×|-2|·|2k| = 3,$即$2|k| = 3,$解得$k = \pm \frac 32$
则$k$的值为$\pm \frac 32$
$(3)k$的取值范围为$1<k\leq 2$或$-2\leq k<-1$