Well, what we first need to figure out is which is more likely to happen;
1) The ice melting AND heating to 100 degrees, and the steam cooling to 100 degrees AND consensing to water.
Or
2) The ice melting and staying 0 degrees and the steam cooling, condensing and cooler even further to zero degrees.
To do that, we do something that is similar to a limiting reaction problem in first semester general chemistry. To do this, we take the extremes of both cases; find the amount of energy required to turn all the ice into water, then heat it up to 100 degrees, as well as how much energy needs to be withdrawn from the steam to cool it to 100 degrees, condense it, and cool it further to 0 degrees.
The math shows that for heating the ice to 100 degrees, it requires ~19,000J whereas to cool the steam to 0 degrees it would require that ~105,000J be withdrawn (hence the -105,000J).
Now, the big concept you need to know; what is causing the heating and the cooling --and the answer is they are causing it to eachother. So the ice can only absorb ~19,000J of energy before it's heated to 100 degrees, but the steam requires that something (the ice) absorb 105,000J of it's own to be brought down to a low enough energy that it be 0 degrees.
So from that we know that there is not enough "energy absorbency" of the ice to condense all of the steam.
Then the rest is plug and chug.
