Almost all of the cooling ice provides is due to melting: the heat capacity of the ice itself (energy release due to temperature change) is much less than the latent heat of fusion (energy release due to melting).

Look at the huge jump at 0 deg C in this diagram: https://upload.wikimedia.org/wikipedia/commons/9/9d/Heat_cap...

It takes less heat to warm ice from -100 C to 0 than it does to melt ice at 0 C to water at 0 C.