Your answer is off by about 5 degrees because you used the wrong specific heat for water. The specific heat of liquid water at standard conditions is about 4.2 J/gK. You used the specific heat of very low pressure steam by the looks of it.
The other Kiwis here have given you proper, mathematical answers, but I'll give you a pragmatic one: honestly, you might want to just fudge it a little if it's for your video game. Realistic heat effects don't always look as cool or as 'intuitive' in video games/animation as just using simple rate equations and linear interpolation.
