This question involves a phase and temperature change, and I am slightly confused on how to go about it. I'm assuming only the latent heat of...

You are right, there is a phase change and temperature change involved in this question. The phase change is for the steam conversion to water. The phase change will release some heat and some more heat would be lost by this water as its temperature decreases. On the other hand, the water (1 kg, to which steam is mixed) will gain this heat and its temperature would increase.

Thus, heat lost during phase change + heat lost by newly formed water = heat gained by water

Assuming that the final temperature of the mix is T degrees C.

Mass of steam x latent heat of vaporization + mass of steam x heat capacity of water x temperature change = mass of water x heat capacity of water x temperature change

Here mass of steam = 100 g = 0.1 kg

Thus, 0.1 x 2256 + 0.1 x 4.186 x (100 - T) = 1 x 4.186 x (T - 10)

or, 225.6 + 41.86 - 0.4186 T = 4.186 T - 41.86

or, 309.32 = 4.6046 T

thus, T = 67.2 degrees C

Heat of fusion is not required, as the heat lost by steam is not sufficient to convert water to ice.

Hope this helps.