Unfortunately a nice pool and free internet are going to be hard to find in central Madrid. I like the Westin Palace for central tourism. Or the Intercontinental if you have Ambassador status.
Taxis are relatively cheap in Spain, so use those for transport. Or the metro if it is convenient. Go see Plaza Mayor, but AVOID the touristy restaurants right in the square. They will try to rip you off one way or the other. The museums are fabulous. Especially the Prado. The kids might be weirded out by Bosch, or they may be fascinated.
