How dependent is the molal depression constant on the nature of solute?
Some solutes form nearly ideal solutions up to moderate molalities, examples being glucose and sucrose in water. Such solutions allow fitting of freezing point temperature data to the following equation:

$$\mathrm{log}(1-x_s)=\frac{\Delta_{fus} H_m}{R}\left(\frac{1}{T_{m}}-\frac{1}{T}\right)$$

Taking various approximations which includes assuming that the heat of fusion is constant from $T$ to $T_m$ (the melting point of the pure solvent), that $x_s&lt;&lt;1$, and that $T_m\approx T$ you obtain the expression for the freezing point depression, $\Delta T = T_m - T$, in terms of the cryoscopic constant $K_f$ and solute molality $m_s$:

$$\Delta T = K_f m_s$$

where 

$$K_f = \frac{M_wRT_m^2}{\Delta_{fus}H_m}$$

Since $K_f$ contains parameters that depend only  on the solvent (not on the particular solute) the equation can (to within the limitations imposed by the above approximations) be applied to any solutes with which the solvent forms ideal solutions. Which is why you can determine $K_f$ with one solute only to use that same constant to later determine the concentration of another solute.