Yes, but melting is just the beginning of the process. Even your computation is incomplete, because it is not enough to heat iron until the melting temperature, you must also provide the additional latent heat of melting. Similarly for boiling iron, after heating to the boiling temperature there is an additional latent heat of vaporization.<p>There is still no easy way to separate platinum-group metals from liquid iron, so you must vaporize the iron, to exploit the fact that platinum-group metals have higher boiling temperatures. It is true however that at the low pressures easily achievable in vacuum, vaporization is easier than on Earth.<p>Otherwise than by vaporization, you could dissolve iron with an acid, but on such asteroids you do not have with what to make an acid, so you would have to transport it from some other asteroid, or more likely from a satellite of Jupiter. You would also need a chemical plant to make the acid and also to recycle the iron salts into regenerated acid. This is so much more complicated, that vaporization of the iron might be simpler.<p>Finally, you must account for the fact that the energy required to vaporize one ton of iron produces less than a gram of platinum and of each other platinum-group metals. It is unlikely that you could build there a solar array big enough to provide energy for vaporizing a million of tons of iron, to make a ton of platinum, so you would need a nuclear reactor.<p>While platinum-group metals might be obtained as a minuscule residue after vaporizing the iron, gold has about the same vapor pressure as the much more abundant iron, nickel, cobalt and germanium, so it would be impossible to extract it from iron by vaporization. It could be extracted only with a chemical method, e.g. with an acid or with oxygen, which need to be brought from elsewhere.<p>Taking all these into account, it seems that there is no chance of being able to mine precious metals at a cost less than on Earth any time soon, e.g. in the current century. Extraordinary reductions in the cost of interplanetary transport would be needed and in the cost of building a metallurgic plant on an asteroid.<p>Mining asteroids would make sense only if some people would decide to live in huge space stations with artificial gravity, instead of on Earth, and then some asteroids would be mined for making steel and other construction materials, to be used in the interplanetary space, not on Earth.