It is difficult to advise you about what apartment choosing, as we all have different preferences and requirements. Choosing the right area is basic for having a happy time in Madrid. For that reason in this Madrid Guide you will find plenty of information about the different Madrid areas, pictures and maps which will help you to take the right decision.
The first aspect you should consider is the location. As mentioned, don't rack your brain, simply because Madrid's 1000s sights are scattered all over the various areas, so you will be roaming around all the areas in any case.
The zone however is the place where *you will get back at night* during the rest hours, so you should consider it the place you would like to have as "your home" in Madrid: where you dine at restaurants after sightseeing, where you have your morning "café con leche" etc.
Second: consider your "budget", or consider what you can really afford. Of course, prices -
Third (for some people it is the first aspect): choose the apartment which you "really like".