Some more comments to throw into the mix.
GreatDane, if you are on Delta, you will not be going into T4. That means that you would take the Cercanías train to T4 and then have to take the intra-airport bus from T4 to T1 or T2. This can add 30-45 minutes onto your journey, turning a half hour trip from Sol to T4 into an hour trip from Sol to T1. But still plenty of time, if you got on a Cercanías train between 7 and 7:30, you´d be fine. There is another option for getting to the airport, which many have recommended, which is the airport bus.
http://www.emtmadrid.es/lineaAeropuerto/horarios.html 40 minutes from downtown to the airport, leaving only from Atocha train station. To use that option, you´d either want to stay near the Atocha station (which has some nice areas, near the Prado museum, the Retiro Park, the Reina Sofía museum, etc.) or take a metro to Atocha from wherever you stayed in Madrid. But that option may complicate things too much, I don´t know how comfortable you are with an option requiring metro and bus.
Liz, I don´t disagree with Olivares´ comments about pickpockets, but they are really well dispersed around all of central Madrid. I think that no matter where you stay in central Madrid, you have to be very sure to have all valuables under your clothes. I have also stayed at Hotel Atlántico on Gran Vía, it´s very quiet because it has double or triple windows. It may be a bit over your price range, probably closer to $200. I agree with Alan that Persal is a good budget choice, not much in the way of charm, but clean, functional, great location.
The last time I was in Madrid I stayed near the Nuevos Ministerios, which gave me a no-transfer straight shot quick ride out to the airport on Cercanías. I stayed in another pretty charmless hotel, the Hotel Zurbano, part of the NH chain, but it was close to a great health club-fitness center, and also in a good neighborhood for restaurants and cafes (I can dig up some names if interested). I had a couple of great meals in the area, and it´s not downtown central Madrid, so it has a more relaxed, less intense feeling. Walking distance to a big Corte Inglés, which I also liked so I could stock up on olive oil, cheese, etc. So all in all, I thought it was a very good option.
And I´ll add my "not to be missed" museum in Madrid -- it´s the Museo Sorolla, Spain´s most illustrious impressionist, housed in his lovely home. The paintings of children on the beaches near Valencia and the other beach scenes are glorious.